Take each of the following steps that you haven't already tried, testing after each one.
Step 1
From the Mail menu bar, select
Window ▹ Connection Doctor
Double-click the problem SMTP (outgoing mail) account in the window that opens. it. Another window will open, showing a list of all the outgoing mail accounts. The one you clicked will be selected.
Select the Advanced tab in the settings for the account and check the box marked
Automatically detect and maintain account settings
if it's not already checked.
If there's more than one problem account, repeat. Close the window and save the changes.
Step 2
Open the SMTP server list again. Make a note of the settings of the problem server, then click the minus-sign button to delete it, and confirm. Click the plus-sign button and recreate the account with the same settings.
Step 3
The mail server may be using a weak, obsolete form of encryption to secure its communications with the clients. The update to OS X 10.10.4 changed the behavior of Mail and other applications so that they no longer tolerate weak security.
Log into the mail account on the website and see whether there is a security setting you can change. I can't be more specific. Refer to the service provider's documentation.
If there is no setting, persuade the server administrator to upgrade the server's security. He or she should do that anyway. Weak security affects all clients, not just Mac users.
Otherwise, you don't have any good options for continuing use Mail with the service. You could disable SSL in the advanced account settings, but that's very unsafe and may not work at all.
You might be able to get a more flexible mail client, such as "Thunderbird," to work with the service, or maybe you can access it via webmail.
Ideally you should switch to a better mail service provider. You wouldn't have this problem with a well-run service. The risks of weak SSL have been known for a long time, and competent server administrators have already taken the simple steps needed to avoid those risks.
