Durable paint baseboards?

Any quality trim paint will be as good as you will get. There is no special paint for baseboards.

you may need to use an oil paint, you might have to remove the paint that’s on the baseboards.

I would try to chip a little piece of paint off and see if the paint store can tell you if it is oil or latex. If oil you can repaint over it with either oil or latex. If latex, you must use latex. I would not go to the trouble of removing the paint. Also I find the way oil paint is mixed now, noxious for indoor use. Just get a high gleam mixture for your latex paint.

A high gloss latex paint is very durable, very washable. Most newer houses have been painted with latex paint rather than oil. Use a quality paint with primer and wash with TSP first.