Several years ago, I painted the back stairs (and the second floor hallway) the same color as the trim (Classic Ivory). That was a mistake. It was much cheaper than getting it refinished, and the original carpet was a disaster, but the paint was impossible to keep looking clean.
So today, I scrubbed the back stairs with TSP, and painted them brown. I think the brown is more traditional and shouldn’t show the dirt as easy. As a side note, the TSP did a great job cleaning off the dirt that had been worn into the stairs, but with two boys and two dogs a light-colored floor is just not a good idea. I’ll eventually repaint the second floor hall, but that is a little bit more involved process (there are two stairs, so painting the back stairs doesn’t impact anyone).
Before and after pictures are below. They do need a second coat, but you can get the idea of how it will look. I used “Floor and Porch” paint from Sherwin-Williams. The paint color is “Tea Chest” (SW 6013).