Parallel Anthology Track 22
Track 22: William and Versey Smith - When That Great Ship Went Down Paramount 12505-B; Recorded in Chicago, August 1927
No image of record label available. Neither Anthology book assigns authorship, nor does www.78discography.com.
The Roud Folksong Index classifies this song as a version of Roud 774. A Roud number search returns a handful of results which predate the Smiths' recording.
According to the 1928 book American Negro Folk Songs, a song with almost identical lyrics was "Sung by a Negro on streets of Hackleburg in Northwest Ala." in 1915 or 1916. 
As this song refers to a definite historical event which took place in 1912, it is quite possible that the person who wrote it lived as late as 1943, meaning that the copyright status of this composition is unknown.
The Titanic Disaster by Pete Seeger album released in 1957 (according to www.folkways.si.edu): Spotify Liner notes treat this song as traditional.
Ship Titanic from Songs of Camp recorded in 1958 (according to liner notes): Spotify Liner notes make no claim for authorship or arrangement.