Measure and cut 2-by-4 inch horizontal studs to fit between the staircase stringers every 16 inches from top to bottom. You'll need to install a stud at the very top end of the stringers and another at the very bottom, then one every 16 inches in between. Use your tape measure to record the space between the stringers, then use your circular saw to cut 2-by-4s to fit. Mark off the uppermost installation location across the top of the stringers, and another at the bottom of the stringers, then every 16 inches on center between the top and bottom plates.
Install one horizontal stud at the top of the stringers. Tilt the wood so the broad side faces forward, and so the front edge is flush with the stringers on both sides. Drive nails through the outside edge of each stringer and into the studs in between. Repeat this process with another horizontal stud at the bottom of the stringers, and again at each pencil mark all the way down.
Measure and cut a piece of 1/2-inch drywall to fit the width and height of your framed out stringers. Use your straight edge to mark the drywall sheet, and use your razor knife to make the cuts. Hold the drywall in place over the wood framework, and use your screw gun and drywall screws to fasten it in place. Install one screw every 16 inches or so around the entire perimeter and across each brace location.