When we bought our house it was new and had builder-grade wall-to-wall carpeting on the stairs.
A few years ago, when we removed the old carpet, we first tried leaving carpet off the stairs. After two days of that I could hardly stand it. My knees were killing me. So, we tried a carpet runner on the stairs. It has a textured pattern that makes it non-skid, more suited for stairs than regular carpet. It helped but it still felt "hard." So, TCS took off the carpet, added a carpet pad to each tread, and then reinstalled the runner. Now it's just right. I can go up and down the stairs all day with no problem. Even with my Parkinson's I've never slipped.
(TCS also added a second matching handrail to our stairwell so that I have one on each side.)