What is a good prevention measure to keep bears from opening your door?
I've trapped quite a few bears near my cabins but not once has a bear ever crossed or broken through a fence, so, build a fence as soon as possible. It's not that they CAN'T get through a fence. They just seem to have a hard time deciding to do so.
Considering the possibility that a bear can open doors, perhaps the fence should not have gates? For myself, I've got baited heavy deadfalls near any gates.
For my cellars, I just put a loop snare on top of them. I've caught lots of birds and hares with those. I've rarely seen tracks of any other animals around my cellars and never repetitively.