I am using my small riser table and a drill chuck on my radial arm saw to drill holes into the ends of the short pieces cut on the bandsaw. Notice that I have clamped a piece of scrap to the top of the riser table. This functions as a fence so I can slide the work into the drill bit. Great care needs to be taken to insure that the holes are as near to centered on the ends as possible. This is so the dowels will align with the holes in the rest of the puzzle and the puzzle pieces will work. During setup I put a very small size drill bit into the chuck and used the scrap pieces to test how close to the center I was. If you look closely at the photo, you can see I marked an "X" on the pieces. The "X" goes down against the table surface when drilling. If there is any error in centering, the error will compliment itself in all of the pieces, if the same orientation is preserved, and not work against itself to multiply as the project progresses.
Drill the holes to a diameter of at least 9/32 inch and a depth of about 1 5/16 inches. Check to be certain the dowels slide in and out of their holes freely and quickly by means of their own weight. Make the holes a little larger, if necessary.
Update: I began to make more of these puzzles than I anticipated and wanted an easier way to drill these holes. I made a special jig
that also frees my saw for other things.