Hold on, let's see what Webster's 1913 says...
2. Skillful in using the hand; dexterous; ready; adroit. "Each is handy in his way." Dryden.
OK, that one seems a bit... vulgar, at worst, and is the sense I took it in, basically.
What I find very funny is sense 4:
4. (Naut.) Easily managed; obedient to the helm; -- said of a vessel.
Might the line have a double-meaning? The obvious one of "knows how to please a woman", but also a near-opposite second reading of "women walk all over him"? Kinda fits with the rest of the song, which reminds me a lot of The Offspring's "Pretty Fly (For a White Guy)".