Hi, Suzanne. If the frame is not split so badly that it looks objectionable from the outside, get some Bondo from the auto department, mix it per label directions with a putty knife, force it into the crack, sand about 100 or 150 grit, and spot paint.
If the frame is cracked so badly that there is an obvious problem when viewed from the outside, you will have to replace one side of the door jamb, mortise for hinges, and repaint. This is not really a big job but it does take a fair amount of measuring and hand tools. You may want to call a finish carpenter who will know how to get the old moldings off with minimal damage and replace the side of the door jamb. If you really want to try this yourself, let us know.