I don't know how you would fix them without paint. You're going to have to put something on to "plate" the metal that's there. It's the previous gold plating that's worn off. There are paint markers available at most craft stores, in both gold and silver. You could cover the area around the earring with painter's tape, and using a marker would give you good control over where the paint goes. If it would get on her vinyl, wipe quickly with a q-tip dipped in rubbing alcohol.
Hope this helps!
Heather