Get used to it and make them clean off their feet. Nothing you can do other than that.

Anything you try to put there would have to be attached to the wood,
which would affect the treads. They make polypropylene tread rugs, but
I'd be afraid the moisture sitting on those would seep down and also
ruin the wood.

Your best bet is to set up a bench at the bottom with a place to
store boots and another to store slippers (to prevent slippery socks on
the wood treads) and require people to remove their snowy boots at the
bottom of the stairs.