AndrewR: the station building itself is closed when the station is unmanned, and just the two side gates on Platform 1 are left open. I don't know if they are left open all night. Presumably when the Platform 2 exit was just a gate, it too could be left unlocked.
Now that Platform 2 has the ticket-shelter-thingy at the bottom of the stairs, Southern are treating it the same as the station building and closing it when the station is unmanned. It seems the Customer Relations Manager at Southern believed that the roller shutter that secures the entrance of the shelter could be remotely operated and therefore could be closed when there were no staff at the station. Unfortunately he was mistaken (because it was installed to TfL's specification, which didn't request remote operation) and therefore it has to be closed by station staff.