There has been a lot of nonsense written about this song - here are some facts and some references to authoritative but opposing articles.

On its first publication in 1889, Yeats said it was "an attempt to reconstruct an old song from three lines imperfectly remembered by an old peasant woman in the village of Ballysodare, Sligo, who often sings them to herself." Yeats' original title, "An Old Song Re-Sung", reflected this; it first appeared as "The Salley Gardens" when reprinted in 1895. In 1909, it was set by Herbert Hughes to the air The Maids of the Mourne Shore. Another vocal setting, by the poet and composer Ivor Gurney, was published in 1938.

There are two views as to which song it was that was re-sung:
There are two views – Michael Yeats in a Lecture in 1965 thought that it came from this, which had been reported by the Dublin City Councillor, Publican and song maker, PJ McCall to have been written down from memory by his father, John McCall (d. 1902) and that he, himself, had heard the song sung by a butcher, James Tierney, a butcher's porter, in Patrick Street, Dublin:

Down by the Salley Gardens my own true love and I did meet;
She passed the Salley Gardens a-tripping with her snow-white feet.
She bid me take love easy, just as the leaves fall from each tree;
But I being young and foolish, with my true love would not agree.

In a field by the river my lovely girl and I did stand,
And leaning on her shoulder I pressed her burning hand.
She bid me take life easy, as the stream flows o'er the weirs;
But I being young and foolish, I parted her that day in tears.

I wish I was in Banagher and my fine girl upon my knee.
And I with money plenty to keep her in good company.
I'd call for liquor of the best with flowing bowls on every side.
Kind fortune ne'er shall daunt me, I am young and the world's wide.

In my view and given that John McCall died in 1902, which gave him had thirteen years in which to construct this from his memory of another old song and his knowledge of Yeats' poem – the first two verses are too little different from Yeats' poem to be its origin rather than derived from it.

The second view is that of Hugh Shields in an article in the Trinity College Dublin Magazine, Hermathena, in 1965. (From 1954, Hugh Shields, a Lecturer in Medieval French at Trinity, collected songs across Ireland, especially in north Derry, and allied them with ballad sheets. His knowledge of the working of tradition was very extensive.) His chosen origin was "The Rambling Boys of Pleasure" a song known in tradition from Robert Cinnamond, Joe Holmes (and other) and widely on ballad sheets (see Bodleian Ballads) - This song includes several of Yeats' lines and a verse saying I wish I was in America which is very like John McCall's verse about Banagher.

If anyone wants the precise references, Michael Yeats' lecture was later published, I can supply them.