Remove the broken drawer from the dresser. Inspect the joint where the drawer is broken, looking for old glue, splintered wood or other sharp edges.
Sand or file away old glue residue and make the surfaces smooth. This will help the new glue to hold better.
Run a bead of wood glue over the broken joint. Use toothpicks to push the glue into the joint. Push the joint together so it is tight again. Use the clamps to hold everything in place while the glue dries. If you have multiple joints on the same drawer that need repair, glue all the joints. Then use a strap to hold all the joints at the same time. These straps may be found at woodworking supply shops as well as home improvement centers.
Let the glue dry. While it is drying, inspect the drawer slide in the dresser. If it is loose, tighten it. This helps the drawer function better and will relieve some of the stress on the joints when the drawer is opened.