Ensure that a closed door is a completely locked door?
Talk to a competent lock dealer and ask about a storeroom lock-set. 

This is intended for secure office store-rooms, where there would never be a reason to leave the door unlocked.  From the inside, the handle always allows egress.  From the outside, the door cannot be opened without a key, period.  There is not push or twist button to make the door stay unlocked.