According to James Revell Carr in the Journal of American Folklore (Spring, 2009, p. 201), "The first commercial recordings of sea music came out in the 1940s, including a collection released in 1941 by the Almanac Singers [which included Woody Guthrie and Pete Seeger].

Stanley Slade's shanty recordings for the BBC were also made in the '40s.

Carr's apparently well-informed statement may not turn out to be definitive, but it seems to be the best anyone can do at the moment.