Parallel Anthology Track 15
Note that the record label in the image shown here is for Conqueror 8010, recorded on 23rd July 1931 , 25 days prior to the Columbia release. No image is available for the record label of the Columbia release which appears on the Anthology. No authorship is assigned on the Conqueror record label, nor by www.78discography.com for either release .
The Roud Folksong Index classifies this song as a version of Roud 2243. A Roud number search  returns two versions of this song collected prior to Crain's recording, but they are both from the 1920s (one from 1921, one from 1927).
Lyle Lofgren says that this song "was probably a poem written for a northern newspaper before 1882, as it says Jesse was still alive. John Lomax collected a version for his 1910 Cowboy Songs. This somewhat different rendition was recorded by Texas cowboy singer Edward L. Crain in 1931."
Harry Smith's Anthology booklet assigns authorship to Crain.
Given the fact that Crain's version is 'somewhat different' to the traditional song, it seems reasonable to assign authorship to him.
The Anthology reissue booklet says that Crain was still alive in 1970, which would put this composition in copyright until at least 1st January 2041.
Cole Younger by Mr. William Edens recorded in 1960: Max Hunter Collection Field recording
Cole Younger by Warde Ford recorded in 1938: Library of Congress Field recording. Performer credits Cole Younger himself as author.