If you are looking to do a decorative carpet on the stairs (only) then one way to do it is to use the carpet as a RUNNER...not a stringer-stringer carpeting. The other thing to consider is normally the stairs should match "something". If you have different carpet at the top of the stairs that does not match the stair carpet, that NORMALLY means you have different carpet DOWN stairs. The stairs will match the carpeting on the MAIN floor. I'm not hearing that this is your scenario. What type of flooring do you have on the main floor? This is what the stairs will be MATCHING if you choose to go this route (stairs not matching carpet upstairs). The other way to do this is to use the decorative carpet on the stairs and then pay to have an area rug cut and bound in your chosen "stair" carpet. This area rug then sits in the entrance/bottom of the stairs. This is a quick, easy (slightly less expensive) way to get the stairs to match "something".....'cause they really should match something. Right now they match the upstairs carpet. Once you change this design feature, you need to REPLACE it with another version of "must match something". The bound area rug in the same carpet is one of the fastest ways to do this. Good luck. Remember to look at your main level flooring and decide whether or not you want to go forward with this concept.

Base or a Shoe moulding?

heritage, I know you didn't . I was just mentioning the unsafe part regarding kids in case the op didn't know. Stairs and balusters are all unsafe when it comes to kids. doesn't matter what they're made of. kids will climb on anything. that's why I mentioned the teaching part. it was nothing against your comment.

I'd go with a very low pile (think commercial carpet) and wrap each tread. Unfortunately, it's going to be expensive. I've priced full runners up stairs and it was costly. Or, you can remove the carpet tack strips and see what shape the treads are in. In the long run, it may be less costly to sand the wood, fill in holes and paint it. You could do black paint if that fits with the decorating scheme or a dark brown that is a good contrast to the wood. Then, you could do just treads on each step. Treads are super inexpensive. Wayfair has a ton of them. Search for rug stair treads. I'd go with a neutral color or a nice neutral pattern. Something that doesn't show dirt. Here's some nice ones wrapped (probably costly) Here's one with treads: I couldn't find an open staircase with treads but this gives you the idea. Imagine this is your painted brown stair tread with carpet. This one is sort of a hybrid of the two styles Really like your staircase and wood ceiling.