Prevent Firefox from storing form entries can help you turn that feature off.

I'd also suggest setting up multiple user accounts on those computers so each person has their own account, which can be password protected

See also:

• Options/Preferences -> Privacy & Security: Forms & Passwords

Thanks Tyler, but we're not quite there. The OP asked about preventing the saving of passwords. You pointed him to a page on storing form entries - but that carries a side-effect that should not, IMO, accompany the solution you offer. The page says: "Disabling form history also prevents Firefox from storing search history for the Search bar in the Navigation Toolbar." Why? That may be well and good, but we here reading are interested in not saving passwords, and passwords alone. It's like giving you an ice cream cone but insisting that you carry a 10 lb weight as well. Why? Is there no way, without other side-effects, of preventing Firefox, globally, from saving passwords?

You probably need a form fill extension if you only want to store the user name. Otherwise you can save a one character fake password in the Password Manager and replace this fake password with the correct password when Firefox has filled the password. You can create a block exception to prevent Firefox from asking to update the saved password.