Which spots is it leaking from? Mostly under/around the door?
The aluminum backed foil duct tape seals pretty well and is reflective for the inside. HomeDepot carries rolls of it in the duct work section. I know more than a few people have commented that flat white paint reflects quite well. Aside from using that or a vinyl film to line the inside I don't know of much else.
I'm a big fan of the gorilla glue waterproofing tape. It's expensive, very sticky, rubberized & bendy, but it's black so not ideal for inside. You could try it on the hinge side as it might flex enough to cover the gap and not break or pull off to easily. Probably be an ongoing battle.