Sounds simple, especially since you're having other work done. You just need a higher wall, probably at least 40" above the front of each step. They can remove the existing "calf-height" wall and frame a higher wall that runs parallel with the stair incline. Drywall finish, then top it off with wood and trim. You can install a second handrail if you like (but not necessary with one on the other side). Do it at the same time as the other wall work and it shouldn't be too expensive.

I just checked our basement stairs. I built a wall that's 36" above the rear part of the step (not the front edge). It looks great and safe. Not sure if it meets code, though.

If you want a really cheap alternative, you could just install a post somewhere near the bottom, then install a rail. But having the wall raised might be best since you will have workers there already.

Thanks for the replies. I did not mention there are no workers, my husband and father in law are doing it. So that is why I am nervous. They do projects well but are not professionals. Can a half wall be a do it yourself?

In my area, heights over 30" require railings or walls. Balusters must be spaced so that a ball 4" diameter will not pass between balusters. Handrails must be placed within certain height limitations. Find out the code for your area and determine if compliance with the recent code is possible for your installation. Do not call the code office to ask, they probably automatically will say that it is required.

Yes, a half wall can be done yourself. Frame it like a regular wall, then attach it to the floor and side of the stairs, then drywall, etc. It'll be solid as a rock.

might it be nice to have the opening in case you need to take the rail off and get something large down the stairs?