the best you can hope for is to use a made for floors urethane or varnish. and there's no guaranty it will hold up as well as on raw wood. but you don't have too much to loose, as striping all that wood is not for the faint of heart.

to do it, you will need to scrub it with a degreaser , then scrub it with ammonia, after that wipe off with a wet rag. it's important to clean the surfaces THROUGHLY . scrub and wipe off with a clean cloth. next use 80-120 grit sand paper and rough sand every inch. then wipe of the dust and you're all set to go.

if you give this a shot , mail me to tank me or curse my name. good luck.

"Any product that can be put over top of the current finish?"
Possibly but I strongly advise hiring a professionel for assessment of what is possible.
This is a classic example of why one should consider all these issues BEFORE deciding to change things.

Short answer is no. There are no good(lasting) products that can be applied to the existing stairs with out stripping the existing finish.

Best method is to strip the stairs, sand with 150 grit sandpaper to smooth and then apply the new stain and finish with a floor finish. Sanding/restaining the hand rail is fairly simple.

The spindles are a pain to strip, so painting them white might be the easier way. Lightly hand sand with 150 grit, primer with a primer like BIN(made by Zinnser) and paint to match the other trim.

Thank you. The house was a foreclosure, so we changed virtually everything. I knew the stairs would be the bigest problem, but I wasn't going to leave all the rest of the trim in the house thee ugly dark original stain, so I thought worst case, I would have to strip the stairs. I do intend to paint the spindles white, so that will be one less area to strip!

Thanks again for your input and advice.