In my opinion, it should just happen. It's not a matter of preferring a girl's way or a guy's way. It's gross to leave it open; the ENTIRE thing should be shut. I hate walking into a bathroom and having an open toilet staring at me.
I understand that I am the only girl in the house with 3 guys (hubby and 2 sons) but still! It only takes a few seconds if even 1 second to put the seat down! I can't stand that my husband is teachign the boys not to put it down! ARG!!