The word is not offensive. Hospitals have Geriatric wards. They are not going out of their way to insult their patients.

I disagree with respect to the term geriatric. It may be a respected branch of medicine, but the term is often used in a derogatory way,........ clearly intended to be offensive.
Well, just about any word can be offensive - it all depends on how it is said. "Gay" is a good example of this.