Is it possible to cure pet dander allergies?

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