Joe Barry replies: Use a V-block to hold the dowel. You can make a quick one by running a piece of wood over the table saw with two 45° cuts meeting to cut the vee. Chuck a 1/16˝ or other small bit in the chuck and align the block so that the bit comes down in the center of the vee. Then secure the V-block to the drill press table so it won’t move.
Richard Oedel replies: Use a V-block (7˝–8˝) which rides against a fence on the bed of the drill press with two clamps, one tight and one loose, and put a drill bit in your drill press. I use a brad point bit so that the hole starts cleanly. Rotate the fence/V-block as a unit to find the center of the fixture by dropping your bit into the bottom of the V-block.
Or you can go back and forth over the top of the dowel bringing down the brad point until you just barely graze the dowel. Tighten the second clamp, and you have a fence that holds the V-block and dowel on the centerline of the drill press! Now just drill and repeat, making sure the dowel does not rotate in the process (hint: double sided tape in the V-block might help).
Buy or make a V-block (shars.com/toolmakers-v-block-clamp).