How do I repair my stair bannister?
Those wooden circles are plugs that will pry out of the holes in the balusters.  Under that would be a pre-drilled hole for a screw (typically).  I would have never used nails on this in the first place.

The correct fix is going to be to remove the nails and replace them with screws that fit into the existing holes snugly.  If there is really just a nail hole, you may want to pre-drill a larger hole (sized appropriately for the screws you choose) and then drive in screws.  An electric driver is going to be the best for this unless you only have a few to repair.  A manual screw driver is going to get tiring and hard to use very quickly.

Once they are screwed into place, the wooden plugs can be tapped (maybe with some glue) back into place.