Mail goes offline by itself. Only restarting Mail helps.
It has become a real drag...
I need to restart Mail every few minutes.
I have many accounts with different servers and providers and they all go offline, so it's not a problem on their side, given that as soon as I restart Mail everything is OK.
I work with email: having to restart always will bring me to abandon Apple Mail after years of faithful usage.
MacBook Pro (13-inch Mid 2010), OS X Mavericks (10.9.4)