It’s easy for inexperienced users to simply close the Mail window and not fully quit the app. This sounds like overkill, but it guarantees Apple Mail has been fully quit and reopened.

If you definitely have internet access, restart your Mac from the Apple menu (top left of screen) > Restart. That way you know that the browser isn’t loading up a previously-saved page and must therefore have internet access. A particularly good test is to perform a Google search for something very random. The easiest way for most users to check their internet connection is to try loading up any website. Apple Mail seems to always assume this is a password problem, instead of checking first whether the internet connection is actually working. This is because it’s unable to make a successful connection to the email service provider.

If your internet connection drops out, even just temporarily, Apple Mail will often ask for a password. But step 6 below fixed it.Strangely enough this is the most common cause of Apple Mail asking repeatedly for a password. Email was still completely functional on my phone and on the AOL website. Everything had been working fine up until it suddenly stopped working. The field for password in the 'Advanced' area of the 'Edit SMPT Server List' was blank. Dear Apple Support personnel: Please check it out and add it to your data base! It may not work for all, but it sure worked for me. Does this really need to be so difficult? The fix seemed very simple. I spent 2 hours with Apple support one day, and a half hour on another day. Pop-ups were popping up all over the place-I would enter the password every time something popped up- entering it in the Accounts Information field in the Preferences for Mail, quitting and restarting Mail, and nothing helped. Pretty jive that you can enter that password in all the other places, and it didn't work.

This was from another site, and it worked.

So, none of the responses from Apple were helpful.

Mail on mac keeps asking for password