The earliest dated reference to the familiar version I've seen is in a description of a "convict camp" at Tracy City, Tenn., published in Salt Lake Herald of March 8, 1889:


“As might be expected in a ["Negro"] convict camp, there are favorite songs about noted bandits, including one about Jesse James, beginning,

        Jesse James was a man
        Who from danger never ran.
        He robbed the railway train:
        But a dirty little coward,
        He shot Mr. Howard,
        And laid Jesse James in his grave.

Chorus – Oh, Jesse had a wife;
        She’s a mourner all her life;
        The children all were brave;
        But the dirty little coward
        That shot Mr. Howard,
        He laid Jesse James in his grave.”