I would paint it black, then place shelving units in front of the railings to hide the railing.

start with painting it black.

then save up for an investment in changing out the railings to something without the finial balls and turned post.

Not exactly what I mean, but you can see the shelving unit

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Start by painting the newel posts, top rail and handrail black. IMO, that will work best with your antiques. Or match those parts of it to your floor finish. Leave the balusters white for a classic look.

Here are a couple stair railings with slanted cables, which avoids the crib-bar feeling. Stain whatever wood elements you include to match the wood floor. Do a Houzz search for "modern indoor staircases." Anything with a lot of spindles is going to give you that crib feeling.

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I would prefer walls to that. Is that a possibility?

I hear it: comments about low walls, and also what for me, might feel like a "pit." Cable design might "disappear' and not 'show' empty spaces [between].

Black has no details:

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My comment above was for walls to the ceiling. Sorry for the confusion.

Might not be a concern, but kids can climb on top a half wall.

Here are some interesting railings.

Right...so you would be creating hallways rather than what you have. Put a door at the entrance to the stairs.