What kind of doctor do I see if I want to get my drooping eyelid corrected?

Droopy eyelid. There are several causes of a droopy eyelid. Your best bet would be to have your eyes evaluated by a board certified facial plastic surgeon.
Oculoplastics. Oculoplastics surgeons specialize in surgeries of the eye lids and orbits. This is the doctor you want to see so long as you already know the cause of you drooping lid.