Seal the base of a drafty wall
If there is a draft, it is 99% chance it is coming from outside.
Uh, yeah you can caulk it. You can caulk anything. I have seen boats with so much caulk on the windows that there was more caulk than window.
If it were me, I would (A) find out how air is getting into the wall and stop that from happening, then (B) remove the baseboard and properly build the wall all the way to the floor using joint compound (drywall wall) or plaster (plaster wall), then reattach the baseboard.
Here is a basic principle: when building walls, the wall is supposed to go COMPLETELY to each of the 4 edges.