With reference to the zipper on the flies of a pair of trousers, the expression zipper problem denotes a man’s habit of sexual promiscuity or infidelity.

This expression has been particularly used in relation to accusations of sexual misconduct against Bill Clinton (born 1946), President of the USA from 1993 to 2001. The following, for example, is from Julian Hall’s Festival diary, about the Edinburgh Festival and the Scottish journalist and author Gavin Esler (1953), published in The Independent (London, England) of Friday 15^th August 2008 [Extra section, page 15, column 1]:

At his Book Festival event, Newsnight’s Gavin Esler spoke of a documentary he made in 1991 about who could defeat George Bush Sr for the Presidency. He suggested to Democratic party bosses that Bill Clinton was their best candidate, an idea met with a prophetic put-down: “What? Governor Zipper Problem?”

Of American-English origin, the expression zipper problem gained prominence when, on Friday 8^th May 1987, Gary Hart (born 1936) announced his withdrawal from the presidential race amid revelations of extramarital affairs—cf. in particular, below, quotation 3.

The texts containing the earliest occurrences of zipper problem that I have found indicate that this expression was originally applied to male Members of Congress, i.e., to men who had been elected to the United States Congress, in Capitol Hill, Washington, District of Columbia.

The author of those texts was the British-born U.S. columnist Diana McLellan (1937-2014), who, in the 1970s and 1980s, wrote the Washington gossip column the Ear.

The earliest occurrences of the expression zipper problem that I have found are as follows, in chronological order:

1-: From Ear on Washington (New York: Arbor House, 1982), by Diana McLellan—as quoted in the Chicago Tribune (Chicago, Illinois) of Sunday 1^st August 1982 [section 12, page 4, column 2]:

Busyness is the reason undemanding young women are so important in the sex lives of many male politicians. The gentleman doesn’t want a lot of scenes, thank you.
If he seeks dalliance, he doesn’t want somebody ratting to his constituents or wife, and in that order. He wants what he wants when he wants it, and he’s got an awful lot on his mind. To ensure that the right thing stays on his mind and the wrong thing remains inside his trousers, a few politicians’ staffs loyally shoulder duties as “the Member’s Zipper Patrol.”
Bottle-blonds are still the preferred trollop type among politicians with zipper problems. Perhaps, like bright birds courting, Hill blonds are simply designed to be instantly recognizable to potential mates. That saves a lot of time. Congressmen are always in a hurry. Besides, other politicians understand the blond syndrome.

2-: From It all comes out in Wash., by Diana McLellan, written for, and published in, the Daily News (New York City, New York) of Sunday 10^th May 1987 [page 32, column 2]:

WASHINGTON—Maybe it’s the Potomac’s water. But something changes in a man when he hits Washington as an elected representative of the people. Suddenly, our hero finds himself at the center of the universe.
He is bathed daily in the warm flattering glow reflected off the marble of those monumental buildings on Capitol Hill. Folks from back home file in respectfully to worship at his shrine, deeply impressed by the power and the glory of the Hill in general, and their Member in particular. He’s surrounded by worshipful staffers, beautiful groupies and fresh young hotshots whose sole purpose is making the boss look good and feel good.
[…]
The Zipper Patrol
Of course he’s tempted to dally. Eventually, many politicians’ staffs here loyally shoulder duties as “The Member’s Zipper Patrol,” in charge of keeping him out of the feathers, or getting him in them as discreetly as possible. Name a pretty young woman in Washington, and you can bet that at some time, at a restaurant, Hill hearing or social occasion, she’s been approached by a gray-suited aide who stoops to murmur, “Senator So-and-So would like you to have a drink with him when he leaves.”
The fact is, with glaringly notable exceptions, politicians are highly sexed. Maybe the hormone that drives them to seek risky liaisons is the very one that drove them into politics in the first place—and the one that made them charismatic enough to be elected. And many, many women, for unaccountable biological reasons, perhaps connected to some primitive urge to mate with winners and enrich the gene pool, simply chase powerful men.
[…]
The sad fact is the Zipper Problem in Washington won’t go away.

3-: From Hart forgot change in his status, by Chris Reidy and James Fisher, of The Orlando Sentinel Washington Bureau, published in The Orlando Sentinel (Orlando, Florida) of Sunday 10^th May 1987 [page A-10, columns 3 & 4]:

The nation’s capital may indeed be an erogenous demilitarized zone, a Brigadoon sanctuary where customary rules of behavior do not obtain.
As Henry Kissinger once put it, power is an aphrodisiac, and nowhere is there more power—nowhere are there more lonely, ambitious men—than in Washington.
“It’s a very old problem,” said Washingtonian magazine’s Diana McLellan, who reigned for many years as the city’s premier newspaper gossip columnist. “We call it the member’s zipper problem.”
The zipper problem, by mutual agreement, is generally kept in the closet. Insiders may whisper about it, make rude jokes about it. But the zipper problem rarely finds its way into the public domain—unless the zipper problem is running for president.
“It’s been going on forever—a conspiracy of silence, an eternal, macho conspiracy of silence,” McLellan said.