There are 2 ways to do this, the first way is in essence just cutting the wall at the top of the skirt board and capping it off and setting the handrail on top of the cap that is finishing off the cut wall, sounds a bit crude, but essentially that describes what you will have.![enter image description here](https://i.stack.imgur.com/50aDg.png)
![enter image description here](https://i.stack.imgur.com/gY9z6.jpg)
The up side to this is you will not need to do a floor repair, depending on where your wall ends where the newel will be.
The other way to do it is by removing the wall entirely, repairing the floor, removing the treads and risers, and set longer ones with return nosings on either side. and re-support the stairs with a new walls under the existing carriages.![enter image description here](https://i.stack.imgur.com/sREfx.png)
![enter image description here](https://i.stack.imgur.com/6kdwk.jpg)
This is the most work, but will give the best result in my opinion.
Jan. 2, 2014 edit
Here is the work I did in 1989 that shows what I did.![Stair1](https://i.stack.imgur.com/kiybG.jpg)
![Stair2](https://i.stack.imgur.com/sfhJj.jpg)
This one shows the best example of the effect I was trying to explain. I do not have a shot that shows how the handrail goes past the ceiling, but with a little imagination you could picture what the rail would have to do if the stairs were much wider at the bottom.![Stair4](https://i.stack.imgur.com/2sAZB.jpg)
![Stair5](https://i.stack.imgur.com/vOgza.jpg)
Edit #2
I did a little calculating to help, if it may![enter image description here](https://i.stack.imgur.com/cpBZs.png)
When concrete is poured so that it will be in contact with a wood part of the building allowances should be employed so that moisture is prevented from becoming trapped and is able to escape quickly. The wood wall that abuts to the concrete stairs should have been isolated with a vapor barrier. The most commonly and easiest to use would be 30 lb. roofing felt tacked along the stair/wall face. Metal flashing and self-sticking flashing would also prevent water from contacting the wall. An isolation or expansion joint would have helped any water that did find a way in to eventually evaporate. Repairing the concrete from contacting the studs is difficult. If there was any space between the surfaces a barrier could be retro-fitted. I'm not sure how feasible it would be to start cutting back the edge of the steps to create a space for repairing the wall? If you could that would enable the installation of treated lumber and water shields (flashing and felt). If not than all that can be done is to silicone caulk the wall/concrete joint, but water will still get in through the concrete by condensing.
Best Answer
You really need another skirt board for the wall. When I have my druthers, I make the skirt board the same thickness as the base that will meet it at the top and bottom so it ties in continuously. If need be, I rip a base so the top element, if it has a profile, will make it look like the base goes right up the wall along with the treads.
As a mention, the material you have for the skirtboard would be used on the outside where the wall is under the staircase, using riser brackets to help hide some of the joinery.