Dresser drawers are simple boxes. They consist of two pieces of 1/2- or 3/4-inch-thick plywood or lumber for the sides, two pieces of 3/4-inch thick lumber or plywood on the front and back, and one piece of 3/4-inch-thick hardwood lumber as a face. The bottom is the weakest point. It's made of a single piece of 1/4-inch plywood or hardboard. If the drawer is overloaded, and gets stuck and you jerk on it or it wears out, the bottom of the drawer falls out, cracks, tilts, binds or collapses. The best way to repair it is by replacing it with a new piece.
Remove the contents from the drawer and take it out of the dresser. Place it upside down on a flat surface. Using a drill/driver, remove the drawer guides from both sides by unscrewing the screws holding the guides to the drawer. Set the screws aside.
Insert the tip of a screwdriver between the bottom of the drawer and the drawer box or frame of the drawer, working from the back of the drawer. Pry up on the bottom to loosen the nails holding the back of the drawer to the box or frame. Pull the nails out with diagonal pliers and discard them.
Tap on the drawer bottom with the tip of a screwdriver at an angle to drive it back out of the drawer. If the front of the drawer is facing away from you, tap the bottom toward your body. When the bottom loosens enough and slips forward far enough, grasp it with your fingers and slide it out of the drawer frame. If it's stubborn, tap on it until it comes out.
Measure the drawer bottom that you just removed. Use a table saw to cut an identical piece of 1/4-inch hardwood plywood to the same measurements. Sand the plywood by hand using 100-grit sandpaper. Round and smooth all the corners and edges. Add stain and lacquer or paint as needed. Allow the finish to dry.
Insert the end of the new drawer front into two dadoes, or slots, in the end of the drawer where the old bottom came out. Tap and push the new bottom into the old drawer. The sides of the new bottom should slide down the slotted channels on both sides of the drawer. The front of the new bottom fits into a dado on the front of the box or frame.
Shoot six 1-inch pin nails, evenly spaced, down through the back edge of the new bottom, 3/8-inch from the back edge. The nails penetrate through the plywood into the edge of the frame or box.
Screw the drawer guides back on using the original screws and screw holes.