It's not different than any other staircase. What railings and how you anchor them depends on how fancy you want to make it.

Since the rest of the area is semi-finished, I'd probably use a 4x4 post with two carriage bolts through the side of the stringer and a 2x4 (6 or 8) from the post to the studs on the wall alongside the stairs. Strengthen the stringers by laying a 2x6 flat on the floor between the stringers at the bottom. Depending on how it lines up, you could attach the railing into the side of the post, use a hanger or just screw it in. Balusters may not be required.

For totally finished basements, here's how I do it:

Worthy. Thanks a lot for answering. Seems like it would be doable although as you will see from the pics we have a metal beam where you assume there is a wall stud "Put a 2x4 (6 or 8) from the post to the studs on the wall alongside the stairs".

you will see from the pics we have a metal beam where you assume there is a wall stud

I'm referring to the wall, which will at some point be framed with studs (vertical).
You might want to run the railing right to the top of the stairs, whether you use dimensional lumber (2x 4 etc) or a handrail on brackets. This house looks like it was built before codes required railings. You'll know their importance once you've taken a spill! Or age.

(My first home had 10 foot ceilings and stairs to the second floor along a wall. No railings. No half wall. I lived there seven years in my sure-footed 20s and thought it looked cool. I always wondered why houseguests walked up hugging the wall.)

Thanks for the clarification. Yes, 1930s house. We are in our early 70s and can still run up and down these stairs but a neighbor fell down his basement stairs and has a brain injury so we thought of ours!