Aging can sometimes lead to an increase in the stiffness at the base of the basilar membrane. what effect would this have on a person's hearing?

The correct answer is "they would lose the ability to hear high-pitched sounds".

The basilar membrane is a structural element of the inner ear which separates two tubes of the cochlea, the scala media and the scala tympani.
It is a critical structure, involved in the hearing sense. It has a complex formation, since it varies in stiffness. More specifically, the basilar membrane is more stiff near the oval window and less stiff near the apex. This variation in stiffness enables this structure to act as a frequency spectrum analyzer.

As people get older, their basilar membrane becomes more stiff and loses its ability to analyze in detail sounds of different pitch and frequency.