Pick a solid color from the tile (?) flooring. Will be more cohesive.

Do you definitely want carpet on stairs and will it be laid in upper hall as well (are you open to a runner).

What is your style, furniture and decor like? Will you be making any other changes to the staircase?

Just based on picture, suggest something modern, low pile (with a Stainmaster application), in a mid range color for practicality, complementary to banister and flooring. It’s a big job, but dark spindles is a suggestion, will feel less broken up.

No carpet needed on stairs.

We definitely want carpet as we have three small kids. need something that will coordinate with the peach marble of our foyer. rest of house is modern/transitional.

I would use a very flat broadloom, like a very short velvet cut pile or sheared loop, like a commercial carpet. I would bind or serge each side and have them attached at the back of each step.

Personally I like carpeting on steps for safety. We had open tread curving stairs in our last place and both slipped and fell on the stairs. Right after we took the carpet off the stairs in this house so the floors could be refinished I slipped and fell down the stairs, and I am literally lucky I didn't break my neck that time.

We don't wear shoes above the first floor. so socks on wood are slippery. Your experience may be different.

What is the flooring going to be next to the stairs since IMO that floor is not attractive and will certainly be the thing that will make choosing the carpet a real pain. IMO that staircase is the star in that entry do the same flooring as you have in the rest of the space and then choose a carpet for the stairs and upstairs hllway that works wit that floor. It appears you are doing a huge reno do not cheap out with the entry flooring.. As for the type and color of carpet I agree a very low tight pile would be best. it looks like you have wood treads so I would just do a runner type instead of the whole tread

I would look into having your treads refinished, at least the ends of the treads. Yours appear worn. As suggested by Patricia Colwell, above, wrapping each tread with a surged flat weave carpet will look modern. You might, also consider refinishing your upstairs hallway flooring and have a runner its length in the same carpet chosen for your stairs. I prefer the look of more wood showing on either end of the treads than is shown in your photo with the existing navy. A nice neutral that coordinates with your tile—a natural oatmeal beige—would like great.