At what age do professors stop taking new students?
There are no hard and fast rules for this. However, in general, the question is not so much time-after-PhD, but rather time before retirement. Most faculty advisors in the final stages of their careers stop taking students, so that they can wind their research groups down gracefully. Frequently, the last years before retirement will be spent writing and teaching, and mentoring a few additional students. The amount of time depends on the average career of a graduate student in the professor's particular discipline, but somewhere between three and seven years before retirement, the "wind down" will begin.