Get used to it and make them clean off their feet. Nothing you can do other than that. Perhaps a carpet runner down the center but that will need to be attached and get crappy in short order.

I’m curious why removing the boots or wiping them off on a rug or towel at the door is not an option?

So your living space is at the bottom of the stairs? Presumably there is a place for boots once you get to the bottom.

Keep an old towel by the boots. As soon as a person takes their boots off, they dry the stairs off with the towel.

I know LL bean makes a product called water hog stair treads. They seem like they would protect the wood as well as perhaps be a safe option to help slipping on the wet wood, snow etc.

$60 for set of four with free shipping. Good reviews too!

https://www.llbean.com/llb/shop/122707?page=everyspace-recycled-waterhog-mat-stair-treads-set-of-four&bc=31-514001-506645&feat=506645-GN3&csp=f