Fill the crack in the door with insulation by spraying expanding foam insulation into the hole. Shake the can, insert the straw into the nozzle, insert the straw into the crack and dispense the foam. Let it set for one hour.
Prepare drywall compound according to the package instructions. Scoop some compound onto a putty knife and spread it into the crack. Work it into any creases that weren't filled by the insulation. Smooth the top so it is flush with the door using the blade of the putty knife.
Leave the drywall compound to dry according to the package instructions -- up to four hours. Sand it with 100-grit sandpaper. Wipe with a tack cloth. Apply two more layers of drywall compound, allowing the compound to dry and sanding it between layers.
Refinish the entire door by applying a primer followed by paint so the patch is completely hidden.