Your first problem is to determine why the door split.
How to repair split wood door.
The customers need to start by detaching all the metal hardware from the door like knobs catcher etc.
Depends on how far the split is from either door edge.
Try to achieve a smooth finish that is even and level with the rest of the door.
I would expect a door from this era has nice big mortise and tenon joints.
Cracks run deep so it is sometimes wise to tape the entire piece inside a plastic bag.
If the split in your door frame is very small you can simply use wood putty to fill the crack.
Tape up the sides and bottom of the piece to keep the resin inside.
The customers also need to remove all the tiny wood pieces from the wooden door frame.
Over the last 70 years the expansion and contraction you are concerned about is most likely what has separated the joints.
Do the repair work before making any cuts or planing on the wood and don t worry about the resin puddling on the outside.
Squeeze or spread a small dollop of wood filler over each scratch on your door.
Cut away the damaged wood along the edge of the door plus maybe 4 5 inches past the damage on either side.
Remove all the metal hardware.
If the split is quite big use wood glue and a clamp to hold the frame in place.
Wood glue of any type aliphatic glues like titebond seem to be popular in the.
Apply a generous coat of wood filler to the scratched area.
Either the wood has dried out or has been wetter than it should have been over some period of time.
Then carefully cut and fit a piece of wood to the recess.
You can then paint or varnish the area afterwards.
Whatever the reason a split or cracked door frame needs to be repaired as soon as possible to avoid security issues.
Use your finger or a putty knife to spread the filler around and press it into the scratches.
Imo putting glue into the crack split and clamping it until it dries is just buying time until it splits again.
Taper the cut on either end so the ends of the cut come smoothly out of edge of the door.