These are architecturally interesting, do you really need to fill them in? If you are sure, start by removing all the carpet (ugh). Then screw wood or plywood to the back of each tread to create the riser- I think 3/4" cdx would be the easiest to work with. Finally I would use 1x2's to create a nailing cleat up each side of the stringer and attach thin luan (finish grade 1/4" plywood) to create the sloped 'ceiling' portion.

I feel the same way about open tread stairs. They make me uncomfortable. HALLETT & Co. gave you the "step by step" solution.

But... you have young children. Perhaps there is a compromise in keeping with the era of your home. Where is Beth?

It’s really just a safety issue and I know I’m not the only one that feels this way. This is a concern of many parents with young kids. For now we put plexiglass in between and that seems to be a suitable fix.

Screw plexiglass panels into the backs. Still maintains the "see-through" effect but is safe for the kids.