I would paint it before - then touch it up after installing it.

Mr. Paint

Yes before,ahhhh... the life of before is so nice No taping,cutting in,......unless you like tedious work do it after.

There is something wrong with that way of painting trim. What about when you caulk the trim to the walls and at the 45's ?
The way to do it is prime trim and install it and then caulk and putty the nail holes. Now you are ready to apply the finish paint and if you get a good angle brush you will not have to tape the walls. Oh you are going to be painting the walls after putting up new trim.
Be sure to use a good acrylic siliconized caulk as crown molding is famous for pulling away from the wall as the wood dries out and just from seasonal changes.

Nail holes???Are you using a framing gun,a a trim nailer you should be able to putty and touch that with an art brush.If you plan on using caulking then you'd paint it after.