I don't think there's any shortcut to regular cleaning and maintenance.
We vacuum our stairs weekly, and steam-clean them around once every six months or so.
Also, make sure you don't use outdoor shoes inside the house - other than messing around with dirt, they can also bring in the germies et al that you don't really need to have to worry about.
But, to attempt an answer, I would probably use smaller scrap cuts of carpet running the entire width of the tread, screwed in to the wood at inner and side edges. Do take this advise with a lot of caveats, however. Little kids running up and down the stairs (and many adults, too), don't necessarily step softly, and after a while it's easy to forget that there's another layer on the stairs which can be dangerous if it ever separates from the base carpet.
A runner may well help, too. In addition to performing the same task as described above, it may be a little more aesthetically pleasing.
Regardless of what you choose, I'm suggesting that you firmly fasten it, noting that regular removal, cleaning and reinstall is a big chore. If not properly fastened, you run the precise risks you're worried about.
Vinyl runner... nope!