Green’s Dictionary of Slang

one who passes counterfeit money, whether cheques or notes.

1858		G. Thompson Gay Girls of N.Y. 116: Jotham was accordingly searched, and upon him was found the pocket book of the ‘dropper,’ stuffed full of counterfeit notes and broken bank bills.
1938		F.D. Sharpe Sharpe of the Flying Squad 297: These are then passed on to other members of the gang, men and women, who are known as ‘droppers.’ Their job is to present the cheques at the banks.
1941		V. Davis Phenomena in Crime 48: The ‘dropper’, who cashes spurious cheques at a bank.
1954		Aberdeen Eve. Exp. 22 Apr. 14/2: Charlie Carstairs was working as a dropper at the Northgate dog races [...] He paid out the notes one by one, carefully dirtied and creased, at tote windows.
1956		‘Charles Raven’ Und. Nights 81: Dropping is dodgy work, so dodgy that the penman, who never drops himself, has to send a minder, known as a topper, to keep an eye on the dropper.
1970		R. Fabian Anatomy of Crime 193: Dropper: Villain who changes forged notes.
1989		J. Morton Lowspeak.