I agree with above poster.

Not enough information.

To help you further, I need photos of all 4 walls of both rooms. What I am seeing so far:

1) The stairs feel MCM and the rest of the trim feels traditional. This feels awkward

2) Support pillars painted and stairs stained feels awkward

3) If possible, I would move the furniture away from the stairs. So, move the table to the opposite wall if possible. Same with sofa. This will work better functionally (foot traffic stays in one area) and visually (furniture not crowded around the stairs).

First that is a 80s staircase no MCM there . I certainly would not close it in that will have a huge effect on how the space feels and IMO not in a good way. Ithink to change the spindles to a nice wrought iron would work with your decor and closer on the treads . I see lots of issues with this space to be honest I think with your style this was totally the wrong house.. I agree to get as much furniture away from the stairs as possible but from what I see there is not much space , even the kitchen has no place for the fridge.

The only real solution is pricey- floor to ceiling glass.

We had the same type of railing in our house, built in '89. I hated it and looked for ways to "fix it" . You can't just pop in wrought iron spindles. The stair skirting and the wood piece that serves as the hand rail are just flat pieces of board and too thin to insert spindles. The entire railing system has to be rebuilt to change to a more traditional look.
I agree that the easiest way to handle this is to have it drywalled in, maybe leaving some open railing at the bottom like these pictures. This would allow you to use both sides for furniture placement, but it definitely closes in the space, so it's a trade off.
These fixes aren't cheap though. It money is an issue right now, just paint everything but the treads white.

The better than only real solution is to wall in each side of the stair, full length and floor to ceiling, and hang the best artwork you can afford on the new walls. Leave the stair risers open. You have furniture up against it so you may as well be honest about it.

I agree with closing off the stairs with wallS. As a temporary solution I would look for a way to close off just the kitchen side, even if it meant cutting a triangular piece of plywood as an insert, and cover it with 2 different wallpapers on each side.

I would move the refrigerator and water unit over to that space and probably put the dining table where the refrigerator is. You might need an electrician to create an oullet for you to plug in the refrigerator and help you create a swag lamp with your chandelier so it hangs over wherever you want to have the table.

But if you can afford to get some real walls built now, that would be best