How do you remove SMTP server "Offline" status?

Rejoice, ye Apple-achians. Here's a Mac-Mini newbie who's coming off many years of Windoze usage.

Mountain Lion/Mail 6.1 (1498) is displaying my SMTP server name with suffix "(Offline)", and outgoing mail fails. How to fix? My ISP's tech support & I spent 45 mins. on the problem and can't for the life of us find anything in "Preferences" or Connection Doctor about it. There is no problem at the ISP's end.

Thanks very much.