Trim separating--caulk or wood putty?

You can get a latex filer made for baseboards and trims that is paintable and that will resist wood expension quite well. You can sand it to get it smooth if you have an uneven surface after applying it. You'll find it in the paint section of most hardware stores.

Presuming that this trim is nailed on like it is in most houses, why not just use a hammer to gently tap it back into place? If you can't see the nail heads and/or you're not good with a hammer (it's easy to miss a finish nail), hold a block of wood against the trim where there's a gap and hit the block (not your fingers).

You may want to get a few extra finish nails (they have a very small head that will nearly disappear once it's driven all the way in) to add some extra grip. If you go this route, you may choose to fill the holes with a bit of wood putty, then repaint. I'd suggest a careful touch up of just the puttied areas, but expect to have to repaint all the trim in case it doesn't come out looking all that great.

This way, you'll fix the problem instead of just hiding it.