If the wood there is not sound then you are likely looking forward to some more serious repair such as replacement of the jamb side board or the whole jamb.
Repairing split door frame hinge.
Scrape the dried glue from the side of the door using the side of the putty knife.
Use wood putty to fill the crack and paint or varnish the door as needed.
Use a clamp with rubber edges for a sturdier grip and also to prevent damaging the door frame.
For a more serious split you may be able to glue and clamp the door.
Take a plain piece of wood the width of your door and cut it down until it fits into the shape left by your former door hinge.
Serious splits in door frames.
How to repair stripped screw holes for a door hinge.
Pull off any small splinters.
Hold until the glue starts to dry.
You don t want to damage the structure of the house.
Carefully remove broken split pieces of wood from bottom of the side jamb.
The hinge mortise is a shallow recess the hinge fits.
Since you say that the wood under the hinge and around the hinge screw holes has split and gone soft it pays to check the wood of the door jamb adjacent to the hinge mount location as well.
Attach the clamp onto the cracked door frame.
Using your 3 8 drill bit drill out the old screw holes.
Once the wood is the right size glue the inward facing side and then press it into place.
The dowel i bought was 48 long so i cut off three pieces that were around 1 5 long.
Drill out the old holes.
Carefully cut the damaged frame away at the marked measurements.
Be careful not to cut too deep into the frame.
This may include covering over any wood stains left by the hinge.
At this point you may need to decide if you will be happy with the patched up door frame when you are done.
Take the trim off the door on both sides using a flat pry bar.
Dry fit your dowels to make.
Apply screws just above and below the marked measurements to help hold the frame tight and help guide the cutting.
Apply carpenter s glue to broken wood pieces and glue them back onto the jamb.