Remove the two pieces of the broken baluster from the banister carefully. Make a pencil mark on the side of each one so you can orient them in the same configuration when you reassemble them.
Drill a hole into each piece of the baluster from the point where it broke. Make the hole approximately 1 inch deep.
Dab glue onto the ends of a 3/8-inch-diameter dowel. Push the dowel into the two holes that you drilled into the two pieces of baluster. Realign the two pencil marks that you made and press the two pieces of baluster together. If necessary, seat the baluster on a solid surface and tap gently on the top with a hammer to get the two pieces completely together.
Replace the baluster in the banister and secure it with two small finish nails at top and bottom.