Of American-English origin, the phrase come up and see my etchings is used as an invitation to sexual dalliance—synonym: come up and see me sometime.

According to Jill Anding in The Times-Picayune (New Orleans, Louisiana) of Saturday 26^th April 2008:

A romantic cliché of the mid-20^th century was developed when the phrase, “Want to come up and see my etchings?” was used in Alfred Hitchcock’s first sound film, “Blackmail,” in 1929 ¹. The phrase is attributed to Stanford White ², a New York architect, who purportedly used the line to induce women whom he wanted to seduce to visit the townhouse that he had furnished with etchings of nudes.

¹ I cannot confirm whether the phrase was used in Blackmail, by the English film director Alfred Hitchcock (1899-1980), as I have not seen the film.
² The architect Stanford White (1853-1906) was murdered by the millionaire Harry Kendall Thaw (1871-1947), because White had allegedly raped Thaw’s wife, the actress Evelyn Nesbit (1884 or 1885-1967), when she was 16 years old. According to Frank Capo and Art & Susan Zuckerman in It Happened In New York City: Remarkable Events That Shaped History (Morris Book Publishing, LLC, 2010), during the trial of Thaw, in 1907, a defence witness “testified that, “Come up to see my etchings” was a line that Stanford White coined to woo girls”. However, I have not found any mention of this in the contemporary accounts of the trial.

These are the earliest occurrences of the phrase that I have found, in chronological order:

1-: From the Daily News (New York City, N.Y.) of Saturday 2^nd January 1932:

A visit to a Broadway studio apartment, supposedly to examine a rare etching, has resulted in a suit for $75,000 against Alexander Bunchuk, orchestra leader.
The action has been brought by Paula Bassaner, a dancer whose last stage appearance was in “The Red Robe.” She accuses the bandmaster of strangling and assaulting her last June 9.
In the complaint drawn up by her attorney, Bernard Hahn of 565 Fifth Ave., she states that she was asked to visit Bunchuk’s apartment in the Capitol Theatre Building to get letters of introduction to foreign producers. Then he showed her into another room to see an etching, she said, and the attack followed, according to her complaint.

2-: From All Wrong, a short story by Aline Morley, published in The Daily Times (Davenport, Iowa) of Wednesday 16^th March 1932—Jack and his wife, Millie, are having cocktails with Mr Tribble, one of Jack’s clients; Millie, “a real old-fashioned girl”, has decided to flirt with Mr Tribble:

Mr Tribble regarded the tiny, blue goblet pensively. “Kind-a, pretty aint it? My wife was a great hand for fancy glassware. She brought a lot home from the World’s Fair. I got it yet”
“You must show it to me sometime,” Milly said with a sidelong glance.
And she had taken it as an insult when Devereaux suggested she come to see his French etchings. She was certainly going the whole way, Jack thought grimly.

3-: From Law Has Evil Mind About Friendships Of Men And Women: Ancient Idea That Being ‘Alone Together’ Convicts Couples Doesn’t Match Modern Social Ideas, by Gretta Palmer, published in The Pittsburgh Press (Pittsburgh, Pennsylvania) of Sunday 9^th July 1933:

If you turn to some of the more old-fashioned etiquet books you will find girls warned that they must never, under any circumstances, go unaccompanied to a man’s “rooms.” The very phrase has a sound of guilt about it. It suggests a discreet Japanese servant, a smell of incense brooding over the place and a bottle of champagne being cooled for the express purpose of bringing out the beast in the host.
Now, any woman of even a very moderate sophistication knows that this is so much melodramatic hooey. She knows that the phrase, “You must come and see my etchings,” may mean exactly what it says and that a man may invite her to supper in his apartment without the trace of a leer. They are, you see, Platonic friends, although the phrase would probably never occur to them.

4-: From the review of Double Harness (1933), a U.S. film starring Ann Harding and William Powell, published in The Kansas City Star (Kansas City, Missouri) of Sunday 13^th August 1933:

Marriage, she [= Joan Colby, played by Ann Harding] believes, is the business of women. Being no cheat, she decides that she will marry some man whom she can benefit. Mr. Powell happens to be the hapless individual she decides is in need of benefiting. He has inherited a steamship line, which makes him rich enough to ignore his responsibilities. The fact he is a playboy who likes to invite pretty ladies up to his apartment to see the view (added note of smartness: he does not invite them to see his etchings) makes him vulnerable in hearts.

5-: From the column Peek-A-Boo, published in The Campus of Oklahoma City University (Oklahoma City, Oklahoma) of Friday 8^th December 1933:

The last report on Miller was, that he was still doing that “hooch” dance of his in the “Goldbug” before a spellbound audience. Why don’t some of you “speeds” go up and see his etchings!

6-: From The Fresno Bee (Fresno, California) of Saturday 27^th July 1935:

Wyatt M. Wayne, amateur artist and art collector, convicted of a morals offense against Miss Marian Palmer, model. […]
[…]
The model had charged Wayne invited her to his apartment to see his etchings and then, beat her when she resisted his advances.

7-: From an advertisement for Companion Pajamas, by Manhattan, sold by Wolf Brothers, “Tampa’s Leading Store for Men”, published in the Tampa Morning Tribune (Tampa, Florida) of Friday 29^th November 1935:

Whether he is the “fire-side-dog-and-my-pipe” variety of man—or the sophisticated “come-up-and-see-my-etchings” type, is the first decision to make in selecting his gift pajamas.

8-: From the column On Broadway, by the gossip columnist Walter Winchell (1897-1972), published in The Morning Post (Camden, New Jersey) of Wednesday 25^th December 1935—the addition of “and can you imagine—he really had some!” indicates that the phrase was already well established:

New York Heartbeat
Sounds About Town: In Brentono’s book store yesterday: “Have you unexpurgated versions of anything?” . . . Midnight Friday on a Bronx Express: “Isn’t that dame in that ad sappy-lookin’?” . . . In a W. 49^th Street one-arm lunchroom Monday at 6:30: “He says I should come up and see his etchings—and can you imagine—he really had some!”

9-: From an advertisement for Sweet Caporal Cigarettes, published in The Gazette (Montreal, Quebec) of Wednesday 15^th January 1936:

“Would you like to see my etchings?”
“. . . Not unless you have some Sweet Caporals, too!”

10-: From Sidetracked, a short story by Jean Johnson, published in several U.S. newspapers on Sunday 2^nd August 1936, for example in The Cincinnati Enquirer (Cincinnati, Ohio)—in this passage, one character explains to the other the meaning of the phrase:

Sally was walking through Central Park the next afternoon when she saw Irving Brandt approaching. Her heart skipped a beat as his eyes lightened with recognition.
“You’re looking very smart today,” he approved. And then he said, “Pardon me for rushing along to my office. I toy with a law practice, you know, in my spare time.”
“I didn’t know,” she murmured, stung by his brisk, casual manner. “I—I’m sorry you haven’t time to talk.” Then, as intuition warned her that she would not see him again unless she herself paved the way, she said, “Maxime tells me that you have a wonderful collection of etchings. She says you occasionally invite people up to see them—” She stopped short as a masklike inexpressiveness stiffened his pleasant features.
“Maxime is a clever girl,” he said slowly. “But not quite smart enough. Good afternoon, Miss Donovan.”
Sally looked after Irving Brandt wistfully. And when he disappeared, she felt lonelier than she had ever been in her life before.
The following evening Sally summoned the courage to telephone him. “Please, may I come to your apartment for a few minutes? I’m so wretched. I’m right around the corner.”
He met her at his apartment door. “You shouldn’t have come,” he scolded. “It’s risky.”
[…]
Exasperated, he strode over to her. “Look here, Sally, must I draw diagrams to make it clear that the invitation to come up and see my etchings is a ribald joke?” He cleared his throat. “A reputation for being ‘hot stuff’ is sometimes harder to live up to than live down, but that’s beside the point. Don’t you see why you shouldn’t have come to my apartment alone? You don’t want people to gossip about you, do you?”

11-: From the review by Herman J. Bernfeld of Champagne Waltz ³, published in The Cincinnati Enquirer (Cincinnati, Ohio) of Wednesday 10^th February 1937:

“Champagne Waltz” has all the sentiment of an operetta to weigh it down. Its humor is derived from such incidents as mistaken identity, gum chewing, dialectal comedy and a fake countess who sells the boys tea sets from the household of the Czar when they come up to see her etchings.

³ Produced by Paramount, Champagne Waltz (1937) starred the U.S. mezzo-soprano opera singer Gladys Swarthout (1900-1969).

12-: From the beginning of an article published in the Great Falls Tribune (Great Falls, Montana) of Friday 12^th March 1937:

What to Do When Gent Orders Pie And Asks for Date; Lesson No. 1 At New School for Waitresses
20 Public School Girls Find Absence of Makeup Helps to Ward Off Invitations to Come Up and Look at Etchings

By Harry Ferguson
United Press Staff Correspondent
Atlantic City, N. J., March 11—(U.P.)—What to do and say when a gent orders lemon meringue pie and then tries to date you up for the evening was the most important lesson on the agenda today when Mrs. Martha Sellers opened her school for waitresses.
A sure-fire formula, it was explained to the 20 pupils, is to keep conversation with the customers to the minimum on the theory that if you don’t open your mouth you won’t say “yes” or even “maybe.”
The 20 public school pupils reported for their class with their cheeks denuded of rouge and the paint taken off their finger nails. That also is part of the plot to keep the customers from getting fresh, because hardly anyone asks a gal with a shiny nose to come up and see the etchings.

On Tuesday 5^th October 1937, several U.S. newspapers—for example The Spokesman-Review (Spokane, Washington)—published the following by the British-born U.S. gossip columnist Sheilah Graham (born Lily Shiel – 1904-1988):

It is now taboo, by order of the Hays ⁴ office to say in pictures—“Come up and see my etchings.”

The column Talking Shop, in The Era (London, England) of Thursday 28^th October 1937, also mentioned the ban of the phrase:

Will Hays ⁴ censorship office has now banned the classic Hollywood line—“Come up and see my etchings.”
But not the classic reply: “May I ’phone my sister? She loves dirty pictures.”

⁴ From 1922 to 1945, William Harrison Hays (1879-1954) was the Chairman of the Motion Picture Producers and Distributors of America (MPPDA); in 1930, the MPPDA adopted the Motion Picture Production Code, also known as the Hays Code, i.e., a set of moral guidelines applied to U.S. motion pictures.

On Tuesday 5^th December 1950, the Daily Mirror (London, England) reported the following:

They claim that Hollywood wolves have changed their line in keeping with America’s Red scare ⁵. They no longer ask a girl up to see their etchings. New line is: “Would you like to come up to my apartment and sign a loyalty oath?”

⁵ Popularly known as McCarthyism, with reference to the Republican senator Joseph Raymond McCarthy (1908-1957), the second Red Scare designates the fear of communism that permeated American politics, culture and society from the late 1940s through the 1950s, during the opening phases of the Cold War with the Soviet Union. Anti-Communist loyalty oaths spread throughout the motion-picture industry during that period—cf. booked any good Reds lately? and Reds under the bed.

The paragraph published in the Daily Mirror of 5^th December 1950 was probably inspired by the following from the column In Hollywood, by Erskine Johnson, published for example in the Dunkirk Evening Observer (Dunkirk, New York) of Wednesday 18^th October 1950:

Hollywoodsman to a babe:
“Would you like to come up to my apartment and sign a loyalty oath?”