There have been many musical versions of the poem, however the first one that I heard and fell in love with was recorded by The Waterboys (which spoken word sections by Tomás Mac Eoin) for their 1988 record album Fisherman's Blues. I really think it captures the heartbreaking otherworldliness of the poem with Mike Scott's haunting vocals matching up perfectly to Mac Eoin's reading of the text.
I also love Loreena McKennitt's version from her 1985 album Elemental.
More recently my brother played me a version by British folk act Merrymouth from their 2012 debut album witch is also extremely good.
I know that there are other song versions of the poem but these are the three I'm most familiar with so sorry if I missed out anything.
The Waterboys - Fisherman Blues Amazon US | Amazon UK
Loreena McKennitt - Elemental Amazon US | Amazon UK
Merrymouth - Simon Fowler's Merrymouth Amazon US | Amazon UK