Green’s Dictionary of Slang

sticky fingers n.

1. a thief.

[US]Monteleone Criminal Sl. (rev. edn) 226: sticky fingers One given to petty theft; to steal.
[US]G. Cuomo Among Thieves 147: The fellow with his eye on the prize. Old sticky fingers.

2. the tendency or desire to steal.

[UK]B. McGhee Cut and Run (1963) 77: I dropped the watch in my pocket. I can’t control my sticky fingers.
[UK]J. Baker Shooting in the Dark (2002) 285: Maybe Ralph’s sticky fingers had upset some kind of drug syndicate and they were looking for revenge.

In compounds

sticky-fingered (adj.)

larcenous.

Pubs Eng. Dial. Soc 62-5 33: STICKY-FINGERED, adj. thievish. ‘He’s a sticky-fingered chap, an’ all. The very fust day he’s out of prison he steals a bag of potatoes out of Widow Baines’ garding’.
[Aus]Sun. Times (Perth) 21 Apr. 1/1: Having picked up the ten pound note [...] he attempted to bluff [...] the name and occupation of the sticky-fingered bounder will appear in due time.
[UK]‘Henry Green’ Caught (2001) 151: When you said about the Regular what ’ad a sister that got took inside for bein’ sticky fingered in one of the big shops.
[SA]Sun. Times (S.Afr.) 6 Jan. 20: The sticky-fingered Mwai Kibaki, who had just added an election to the list of stolen goods in his possession.

In phrases

play sticky fingers (v.)

to rob.

[US](con. 1960s) D. Goines Black Gangster (1991) 133: Find out if Roman is playing sticky fingers with our bread.