I would add some type of carpet to the stairs, a dog on wood stairs is asking for a hurt dog IMHO

I agree, that’s what worries me but if we do the wood in both rooms would you still carpet thr Steps?

Some dogs have no problem using wood stairs. If you are worried about yours, a runner (on the stairs) between floors is perfectly acceptable,

I would just install a stair runner on the 5 steps. Looks nice and problem solved.

he is crazy and doesn't know how to walk he runs and he's already had one knee fixed just wasn't sure if the rug on the steps only would look ok. thank you!

A runner on the steps will look fine. Since he's 'crazy' I'd also put a washable, latex-backed rug at the top and bottom of the stairs.

You can also add just strips of carpet on each stair as shown here.

that I will do for sure! thank you

annaztoo nailed it,