I thought diphtheria had been wiped out a long time ago, but my mother says that the vaccinations are still being given. Is it necessary for babies to be vaccinated?

Yes. As we all emphasize, the only truly eradicated vaccine preventable disease appears to be smallpox. Although some are rare, there are pockets of disease thruout the world. All it takes is one infected traveler interacting with a susceptible population to trigger a serious outbreak.
Yes. Diphtheria is a serious disease that still occurs in other countries (meaning it is one plane ride away from us). In america, kids get vaccinated against diphtheria, and adults every 10 years with their tetanus-diphtheria boosters. So, we don't see the disease here. However, if the vaccine is stopped, the disease will be right back.
It is not eradicated. I guarantee you that if children stopped being immunized against the disease, children would start dying, not just being sick, from diphtheria. It is not a pleasant way to die.