I removed all the door trim and applied GE Gray Polyurethane caulk, then reinstalled.

Be sure to keep it away from the wood. I only caulked the outer edge of the inner trim piece. I used acetone and a lot rags and elbow grease to remove the excess. Be sure and caulk this area generously, you do not want leaks here....esp the outer trim piece.
Caulking the Door Jam Trim