Well, you COULD clean off the old carpet adhesive (if the carpet is glued down) and just paint the steps. That would be the least expensive treatment, but would probably look kinda rough tho. If you were to do that, you'd want to use an oil based paint that dried to a very hard film, like a pre-tinted alkyd or polyurethane FLOOR paint. If you can't find a floor paint in a colour you like, you could opt to use an alkyd wall paint, but it simply won't stand up as well as a floor paint because it will dry to a softer film. The softer the film, the sooner it will show wear, and the more easily damaged it will be.
Basically, you can put ANY kind of flooring on your steps. They sell stair treads in metal for ceramic tiled concrete steps, you can remove the old carpet and put in new carpet. Or, you can install vinyl or rubber stair nosings, and then put a wide variety of floorings on the stair treads themselves.
For example, take a look at the vinyl stair nosing from Johnsonite in this picture on the left:
Johnsonite Old > Finishes & Accessories > Stairwell Management > Nosings
The nosing gets glued down to the stair tread with a suitable flooring adhesive. There is a short lip 1/4 of an inch long on the nosing in which you can slip a vinyl, rubber or vinyl composition floor tile cut to fit the depth of the step. As long as the flooring is 1/8 of an inch thick, the cut edge of the flooring will slip under the lip of the stair tread to provide an attractive appearance. A thick (and therefore strong) real linoleum (like Marmoleum) would fit in that 1/8 inch slot as well.
You can get similar vinyl stair tread nosings with a 1/4 inch wide slot in them so that carpeting can be glued down to the treads and the cut edge hidden under a lip on the stair tread nosing.
The stair nosing on the right has a similar feature, but it's made from synthetic rubber rather than vinyl. Both will give you an attractive stair case, but in a house, you really don't need the durability of synthetic rubber. The flooring you put on the steps will wear out long before even the vinyl nosings do. Installing synthetic rubber stair tread moldings would be a waste of money because you'd be pulling off the flooring long before the stair nosings were near the end of their life.
Johnsonite is the biggest name in vinyl and rubber flooring and flooring accessories, but they're not the only ones making such products. Roppe and Bengard also make vinyl flooring accessories like stair nosings.