Watch this video from This Old House to learn how to repair a baluster.
1. Set one half of busted baluster in routing jig.
2. Use plunge router to cut a 3/8-inch-diameter by 1 ½-inch-deep dowel hole into the broken end of the baluster.
3. Remove first half of baluster from the jig, insert the second half and rout the mating dowel hole.
4. Squeeze glue into the holes in the balusters.
5. Insert the hardwood dowel halfway into one of the holes.
6. Push the baluster halves tightly together, making sure the parts are properly aligned.
7. Install baluster in staircase balustrade, and secure it with 8d finishing nails.