Wordorigins.org

This slang term for the police dates to 1924. The origin is unknown. From an article about pickpockets in the Los Angeles Times of 30 January of that year:

A “mob” can “beat a pap” to the “leather” and get away with it with the ordinary “fuzz” lookin’ on.

While the origin is unknown, one 1931 source, Godfrey Irwin’s American Tramp and Underworld Slang, proffers the following:

Fuzz, a detective; a prison guard or turnkey. Here it is likely that “fuzz” was originally “fuss,” one hard to please or over-particular.

Explanations that the term stems from Fuzzy Wuzzy the poetic bear or, bear, the slang term for a policeman are incorrect. This slang sense of Bear does not appear until 1975 and is a reference to the “Smokey the Bear” hats that state troopers often wear.

(Sources: Oxford English Dictionary, 2nd Edition; Historical Dictionary of American Slang; ADS-L)