It sounds to me like you're using a hollow core interior
door as an outside door that is exposed to the elements.
What is at the bottom of this stairwell? Is there a good
quality insulated door with proper insulation all around?
What about the stairwell going up to your living area?
Is it completely enclosed with studded walls, fiberglass
insulation, sheathing, vapor barrier, drywall, etc?
It doesn't really say much in your question about the
stairwell. From the sounds of this, you definitely need
a good exterior door and a properly insulated door
jamb/frame. An interior door in this sort of situation
is not going to cut it. You're going to have to spend some
money to make your home weather tight.