Green’s Dictionary of Slang

(US) of a bank, its money or financial affairs generally, unstable, dubious etc; thus extended to gamling contexts (see cite 1963).

1840		Morning Herald (N.Y.) 27 Mar. 1/5: [A man asked if he could get change for a ten dollar bill] ‘City money?’ asked the cautious clerk. ‘Oh, certainly,’ said the gentleman. ‘Do you think I should ask you to change any red dog stuff?’.
1848		Bartlett Dict. Americanisms 272: red dog money, a term applied, in the State of New York, to certain bank-notes which have on their back a large red stamp.
1862		N.-Y. Trib. 14 June 4/2: We have suffered from red-dog bankers and one-horse brokers for the last twenty years.
1877		Stark Co. Democrat (Canton, OH) 15 July 1/2: They are opposed to all wild cat, red dog, rag baby money schemes of issuing paper dollars without limit.
1912		Literary Digest 5 Oct. 548: National banks [were created] [...] after the era of wildcat and red-dog private-banknote currency [DA].
1915		Day Book (Chicago) 23 June 10/2: A great city like Chicago has to make a bonded debt and borrow [...] is worse than the wild cat, red dog banking system of the ’50s of the 19th century.
1963		News (Paterson, NJ) 3 July 42/1-2: He was inveigled into a game by a red-dog mob [...] When he got into a red-dog game he recognised the danger signals.