Remove the drawer, and place it on a solid work table.
Mark evenly spaced spots, approximately 2 inches apart, along the end of the butt joint at the corner. Mark the corner along the side of the drawer, not along the exposed or cut end. The marks denote where you will drill holes that pass through the side of the box at the corner and into the end of the glued, perpendicular side of the box.
Select a drill bit that is the same diameter or slightly larger than the diameter of the dowel pins or rods. Fasten the bit to the drill.
Drill straight through each mark at the corner. The depth should equal the length of the dowel pins. Drill approximately 2 inches deep if you are using a dowel rod.
Coat the dowel pins or the end of the dowel rod with wood glue.
Tap each pin into a drilled hole with a hammer. If you are using a dowel rod, push the glued end into a drilled hole, and cut off the excess with a hand saw or a sharp utility knife. Repeat at each drilled hole.
Wipe off excess glue with a rag, and let the repair dry overnight.