That completely depends on your controller, and if it requires any drivers to function properly..
Windows simply detects a connect device and responds to whatever the controller says the button assignments are. Most of us use PlayStation 1 or 2 controllers via a Playstation USB adatper. I've never heard of or had problems with the right analog stick working instead of the left, though.
What controller are you using? I don't know if it will help us help you any more than we can already, but am interested nonetheless.
Also, when it comes to Sega Genesis games (or any system Fusion emulates) I don't really know of any that would benefit more from an anlog stick, versus just using the D-pad.. Do your D-pad and buttons at least work properly?