Troubleshooting Apple Mail
What does Mail/Window/Connection Doctor Show? If the server is red, select it and look at the Show Details box.
Mail: Mail troubleshooting
Mail: If you can’t receive messages.
Mail: If you can’t send messages.
Troubleshooting sending email messages
Mail not sending or receiving messages after updating OS X.
SMTP servers keep going offline
As a backup, if you have emails in your account inbox, select the mailbox, then Mail/Mailbox/Export Mailbox, and export the mailbox to your Desktop. Then go to Mail/Preferences/Accounts, select the account, and delete it using the minus sign. Quit/reopen Mail and add the account back. Test. If necessary, to recover missing emails, use Mail/File/Import to import the mailbox on your Desktop. It normal shows in the On My Mac section. Drag emails to the account inbox if desired.