The British- and Irish-English phrase see that’s wet, see that’s dry, and variants, are used to emphasise the truthfulness and sincerity of what one is saying.
—Cf. also cross my heart (and hope to die).

The phrase see that’s wet, see that’s dry, and variants, derive from a children’s oath described by the English folklorists Iona Opie (1923-2017) and Peter Opie (1918-1982) in The Lore and Language of Schoolchildren (Oxford University Press, 1959):

Code of Oral Legislation
[…]
Cut my throat. Of the oaths current today this is the one which was the most documented in the nineteenth century, perhaps because it is the most dramatic; and it is general throughout the English-speaking world. It is another instance in which the special properties of spittle is [sic] recognized. A child moistens his finger and shows it, and says:
          ‘My finger’s wet.’
He wipes it—usually in his armpit—and shows it, and says:
          ‘My finger’s dry.’
He tilts his head back, draws his finger across his throat, and says:
          ‘Cut my throat if I tell a lie.’
Only the most depraved will tell an untruth after repeating such a formula, which is made not a whit the less startling by the miniature stature of its practitioners. The wording varies only slightly from one place to the other.

G. F. Northall had already described the oath in English Folk-Rhymes: A Collection of Traditional Verses Relating to Places and Persons, Customs, Superstitions, etc. (London: Kegan Paul, Trench, Trübner & Co., Ltd., 1892):

Miscellaneous Nominies.
Moral Code of Children.
[…]
          “Handy bandy, sugar candy,
           Cut my throat, and double hang me,
           Job! Job! Job! at ten o’clock at night.”
A rhyme common in the North Midlands, by which one assures a companion of the truth of a statement he makes, or the performance of some act or promise. To “job” is to give a prod or uncertain blow with some instrument.
At Smethwich, county Stafford, the rhyme goes—
           Hangy Bangy cut my throat,
           At ten o’clock at night;
           Hang me up, hang me down,
           Hang me all about the town.
believing if they do not perform according to promise, the spirit invoked will certainly appear and cut their throats.
Another form of assurement is to wet the forefinger, dry it again, and cross the throat, at the same time saying—
          “See that wet, see that dry,
           Cross my throat before I die.
           Cut my throat before I lie.”

These are, in chronological order, the earliest occurrence of the phrase see that’s wet, see that’s dry, and variants, that I have found:

1-: From Summer Gooseberries *. Hairy Ones. Broad in the Chest—and Fishy, published in The Belfast Weekly Telegraph (Belfast, County Antrim, Ireland) of Saturday 6^th July 1889:

He […] proceeded to show me several letters which had appeared in the newspapers intimating that a strange and wonderfully musical bird, which sung only at night, had been heard at Ballyclare. “Anything strange in that?” I asked. He answered promptly that there was. The bird had first been heard by Mr. Baird, of the Hotel, Ballyclare. Its song was that of the nightingale, long, continued, full in tone, complete in register. “But.” he added, “this thing cannot be, for the Encyclopedia says that there are no nightingales in Ireland, and yet it must be a nightingale—if it is.” I saw it was necessary to proceed cautiously, and asked him on his honour if there was anything political in the matter. Was it a Plant? On the affidavit of “See, that’s wet; see, that’s dry,” and the subsequent understanding to cut his throat if he deviated from the truth, he told me there was nothing of a political tinge, direct or indirect, open or concealed. I immediately concluded it was a big gooseberry, and a hairy one at that, of the Silly Season.

* The phrase gooseberry season, denoting the time of the year when gooseberries are ripe, has been used, especially in big gooseberry season, to denote the summer months, when the newspapers cover trivial material because of a lack of more important news.—Synonym: silly season.

2-: From the above-quoted English Folk-Rhymes (1892), by G. F. Northall.

3-: From “And a Little Child Shall Lead Them”, a short story by A. Neil Lyons, published in The Clarion (London, England) of Friday 20^th March 1903:

Then Robert and his ally took up the chase again, and, after an exhausting and exhaustive search, happened upon the entrance of an alleyway where, surrounded by an assortment of sweethearts, stood the Captain. At the moment of surprise, the Captain was engaged in addressing some observations to a small boy in sticking plaster, the only other gentleman of the party.
“If you so much as blow your breaf towards ’er,” the Captain was saying, “you’ll know the results. That’s all. You may go.”
“Thank you, Captain,” responded the object of this address. “See this wet, see this dry: so perish me pink if I do lie. I can’t say no fairer than that, and so you can take it for granted that my words is true. I’m done with ’er for ever.”

4-: From Can English Girls Kiss?, by ‘F. de Plimsolle’, published in The Daily Herald (London, England) of Wednesday 5^th April 1911:

“Clarice,” said I, suddenly, between the acts last night, […] ‘Can English Girls Kiss?’”
“When they go shopping? Why, of course not. The idea!”
“I said nothing about shopping. But can they? […]”
“Why should they?” inquired Clarice, innocently, turning on the fatal baby stare.
“Upon my word,” I hastened to say, “I never really thought of that. Why, indeed? […]”
“Now look here, Frank. You’re a dear boy, and all that; but your outlook on life and things—”
“Meaning kisses?” I hazarded.
“Is far too materialistic,” she went on, ignoring the unseemly interruption. “You seem to think that kisses are in the same category with golf, and hockey, and bridge.”
“Oh, no; perish the thought. See this wet, see this dry—cut my—”
“Cut your nonsense, sir. In one word, desist!” And when Clarice says “Desist!” she means it—sometimes.