"Drunken Sailor" may be more likely to derive from the Irish tune than the other way round, if only because the shanty tune is simpler.
(Before somebody else points this out, the adaptation *could* have gone either way. But folk tunes, like folk lyrics, tend to wear down after long use.) Also, many shanty tunes appear to have come from prior sources: "Donkey Riding" (from "My Bonnie Hieland Laddie"), "The Hog-Eye Man" (from a minstrel song), "The Harp Without a Crown" (from "The Wearing of the Green"), "New York Girls" (from "Larry Doolin"), etc.

Not surprisingly, many of these melodic adaptations are from Anglo-Irish folk/pop culture; many first- and second-generation Irishmen served before (and abaft) the mast in British and American vessels.