Just be aware that if you use a solvent based (oil-based) paint on the stairs it will take a while to fully harden so you'll need to avoid walking on the stairs for a couple of weeks to avoid spoiling the finish.
You'll also need to take this into account when applying a protective coating and, also, the potential that it may not be compatible with the paint you've chosen. In which case, you are advised to do a small test area first just in case there is a problem.
Obviously, it may not be practical to avoid using the stairs at all for such a length of time but the more you can plan around this issue the better.