Pull the broken dowel out of the dowel hole. If this is not possible with either pliers or a pin puller, you will need to drill out the hole with a drill bit. Start with a small bit and move up incrementally until the entire hole is clean.
Remove any wood glue that is left around the dowel hole with a wood chisel.
Squeeze wood glue into the dowel hole and work it around the inside of the hole with the back end of a pencil.
Cover the lower end of the dowel with wood glue. Work the glue into the grooves with your finger.
Place the glued end of the dowel into the dowel hole and tap it into place with a rubber mallet.