From the notes to the Penguin Book (1959):

"There is a common superstition, older than Jonah, that the presence of a wrongdoer aboard ship may make the vessel unmanageable.¹  Disaster may result unless the offender is discovered and thrown overboard.  A Scottish text printed in 1827 makes it clear that the "Jonah" motive lies within this song, though the 20th. century versions are so disordered that the meaning is rather obscured.  The full story concerns a young woman who robs her parents, at her lover's request, and sails away with him.  During a storm at sea the woman gives birth to a baby.  The sailors fear that someone aboard is flying from retribution.  The blame is fixed on the woman, and to her lover's grief she is thrown overboard.  Later versions, however, make it seem that the lover is the murderer.  Fifty years ago Sharp reported the song "very generally sung throughout Somerset".  Five of Sharp's nine Somerset versions are given in FSJ vol.II [issue 6] pp.33-6, and FSJ vol.III [issue 13] p.292 has a Hampshire version² noted by R. Vaughan Williams."  -R.V.W./A.L.L.

This version was collected by Cecil Sharp from Mrs. Overd of Langport, Somerset, in 1904, and was first published in the Folk Song Journal, vol.II [issue 6] p.34.

Other versions on the DT:

Banks of Green Willow  transcribed from a record by Frankie Armstrong, with tune.

The Banks of Green Willow  transcribed from a record by Nic Jones; no tune is given.
Bonnie Annie  from Folk Songs and Ballads of Scotland, ed. Ewan MacColl, with tune.

In the Forum:

Banks of Green Willow A brief discussion of the significance of the title.

Child #24
@love @bastard @sea @murder @deadbaby @ship @Jonah @death

There is an entry at  The Traditional Ballad Index:
Bonnie Annie

Also called The High (or, Green) Banks of Yarrow.

In The Constant Lovers (E.F.D.S. Publications, 1972), Frank Purslow gives a version with text collated from several collected by George Gardiner and the Hammond brothers, and comments:

"In some versions the crew cast lots to discover who the wrong-doer is; in others the girl admits her own guilt, as here.  Older sets of words describe the captain writing a letter to friends ashore asking that, if her body is found, she be buried on "the high banks of Yarrow".  By the end of the 19th. century this phrase had apparantly become corrupted to "by the banks of green willow", probably by country singers who had no knowledge of Yarrow, and it is under this title that the song appears on 19th. century broadsides."

² Vaughan Williams' 1909 Phonograph recording of Banks of Green Willow, probably sung by David Clements, is available on  A Century of Song: A Celebration of Traditional Singers Since 1898  (EFDSS CD02, 1998)  The tune is the one used by Martin Carthy for his 1972 recording of the song.

¹ Another song based around this motif is The Guilty Sea Captain (Laws K22, DT #563), of which there are two versions on the DT:

William Glen
The New York Trader

See also the version from the Penguin book:  The New York Trader

Arguably related to these is  Brown Robyn's Confession  (Child #57)

Malcolm