Is it possible to cure pet dander allergies?

Yes: With allergy shots, you can reduce the severity of cat or dog allergies. This usually improves but does not always completely eliminate the sensitivity. Some precautions in the home, like a hepa filter, vacuuming with a filtered vacuum, polished floors and bathing the pet twice weekly can significantly reduce the amount of allergen around, but may be difficult to do.

Partially: There no "cures" for allergies, but allergy shots, when properly prescribed by an allergist can reduce sensitivity, often to a large extent.