I strongly prefer them closed, but sometimes I have to do both if the bag is so full it won't close or I just happen to only have a single clasp or small flap closure, which is what at least half of my bags have.
when I do leave my purses open I'm always watching-- its not as tiring as it sounds, I generally keep my bags close to me anyways because I do not like other people brushing up against me (I've noticed this, even if there's plenty of room--those ladies really bug me at the moment when it happens) because I don't want anyone touching (re:dirtying) my bag..
but most of my bags are of snap closure, turnlock, or flap..I don't like zippers much because sometimes I fill it to the brim and I don't like for it to get caught