When you say "caulk" are you referring to wood putty or actual caulk (like in a tube and flexible) ?
The joints in crown should be very close so that only a small amount of wood putty fills any gap or imperfections, then after it is dried it is hand sanded with a small piece of sandpaper. I say hand sanded as it is much easier for most people to use and ensure that they don't sand too much and change the crown molding profile.
If flexible caulk was used it's impossible to sand it and it will shred when sanded. In this case it will be much easier to remove the caulk and then re apply wood putty.
As for the application, your mother and father gave you the best tool known.....your fingers , always apply in the direction that pushes the putty into the joint, you can leave a little extra on the joint to allow for shrinkage or just apply a second application if required. Whatever you do do not leave a lot on there as sanding a big build up of putty is a pain in the butt.
I prefer Min wax putty as it dries quickly and accepts stains and paint very well, but there are many good wood putty's available, look for one that mentions it has wood in the putty.