Wordorigins.org

This slang term for the police dates to 1929. The origin is unknown. From Ernest Booth’s Stealing Through Life of that year:

Don’t run, and rank yourself—the fuzz don’t know what’s doin’ yet.

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)