Clarifications, please. A tilde next to the In Box name (e-mail address) in the sidebar of Apple Mail? Where are you seeing this tilde?
Check your "Accounts" settings in Mail Preferences?
probably seeing the "offline" zigzag beside the inbox.
mouse over the zigzag and click to "take all accounts online" and see if that helps.
quit and relaunch mai and repeat if not.
if it still goes back to offline, try running the connection doctor in the menubar under the Window menu, tell us what that reports.