Nails sticking out of balusters
18 gauge trim gun nails don't take well to nail sets, especially when they protrude that far. They bend too easily and you'll find yourself with more damage to the wood.

Grab them with a needle-nose pliers and bend them back and forth a few times. They'll break off. If you do this well they'll break below the surface and you can putty the holes. If not, now use a nail set to push the resulting stub in, then putty. 

If you're looking for a stabilizing repair, I'd drill 3/8" horizontal holes 3/4" deep about 1" up from the bottom on one or more sides. I'd then pilot inside those holes for trim screws at a steep downward angle, into the floor. I'd then glue in 3/8" wood buttons to cap the larger holes. You could also use flush plugs.