That's called the skirt, not the stringer. The stringer is what's under the stairs supporting the treads.
If you can find a stud you can predrill and counter sink holes in the skirt and use ceramic coated 2-1/2 decking screws. to pull it tight.
You can also add base cap moulding to give it much better finished look. http://www.woodworksofpeoria.com/base-cap-molding
If these stairs are wall to wall and not open one side it would do more harm to pull the stringer/skirt to the wall especially if it is a mortise stairs. If it is open one side then there shouldn't be any problems pulling the stairs to the wall provided you pull the entire stairs not just the stringer/skirt.
You will have to push the stairs back against the wall before installing screws, screws alone will not pull that entire stair case.
I think countersinking with the screw will work, just do not pull the skirt trim board in with the screw to far. Drill the screw slowly so that it does not pull the skirt trim up to tightly to the wall. A good grab into the wall stud with screws to pull the trim in some would help keep the trim from pulling away from the wall.
Tip- A simple way to find a wall stud is to tap the butt end of a hammer against the wall and listen for the solid sound, it will sound more hollow between the studs.
Could you post a picture standing back at the bottom of the stairs showing both sides in the same shot?
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