Green’s Dictionary of Slang

(US Und.) a pickpocket who specializes in removing small change from the victim’s fob pocket; Irwin, American Tramp and Und. Sl. (1931), suggests that such a pickpocket has lost his skills, and can no longer attempt less accessible pockets.

1931		Irwin Amer. Tramp and Und. Sl. 78: Fobber.–An old pickpocket, or one who has lost his dexterity and cunning, and is able only to steal from outside or fob pockets.
2000		Indep. on Sun. Rev. 23 Jan. 24: If you can’t tell a slicker from a fobber.