Use wood putty to fill the crack and paint or varnish the door as needed.
Split wood door hinge.
Serious splits in door frames.
Wood glue of any type aliphatic glues like titebond seem to be popular in the.
For a more serious split you may be able to glue and clamp the door.
This video shows how to glue a cracked door stile back in place.
Cracks on doors are everyday problems.
I m thinking the best way to fix this short of ripping out the whole door frame would be to drill out the split wood chisel it square then glue in a new piece of wood for the hinge to be screwed into.
A minor split may be cosmetic.
At this point you may need to decide if you will be happy with the patched up door frame when you are done.
In particular the side where the hinges fasten onto the door tends to crack or split in certain situations.
The top hinge on one door has come away from the door frame the wood is split and soft so the screws have no bite.
It may have holes for screw mounting or spots for welding depending on the style of the hinge and the door material it is mounting to.