Would like suggestions on finishing stairs going down basement.

It might be a bit scary under the carpet, but it could definitely be worth it to remove and brighten up the look- https://www.hometalk.com/diy/staircases/stairway-to-heaven-we-removed-our-old-stained-carpet-and-updated-with-paint-pattern-442082

Here is a link from Pinterest. They have lots of ideas. Even if things don't look 'great' under the carpet, you can still clean it up, sand it and paint it. Some people have even covered over stairs with new wood slats, and then paint or stain:

https://www.pinterest.com/explore/basement-steps/?lp=true

First take up the old carpet to see what you are working with. If the wood underneath is in good condition you have lots of options- you could stain and seal the steps, paint them, or even recarpet them. If they are not in good condition you will probably have to tear them out and replace them. I have seen stairs where the risers were tiled, painted and even stenciled! If the actual step is in good condition you are already 1/2 way there! Just make sure you use a good poly paint and do several thin coats to make sure it lasts. If staining the steps a good poly dealer is the best way to go... either in gloss, semi-gloss or satin. Flat won't look as good or stand up to a lot of foot traffic. I painted my risers in an hombre lok- the bottom riser is black and the color lightens to a soft grey at the top(7 steps/risers)... the walls are a soft grey and the paneling is black, as well as the fireplace mantle. The fireplace bricks are a slightly darker grey than the walls. The floor is a vinyl black marble with white veins so it all coordinates. For a pop of color I have a bright red shag carpet in front of the fireplace. The room has a huge sliding glass door so it isn't dark or gloomy- I put up black " black-out" curtains so there wouldn't be a glare when watching tv/ movies. Have fun and get creative!!

I'm doing the same and plan to paint. I plan to use good rubber treads for safety.