Hi there tomnseattle,
Welcome to Apple Support Communities. The steps to help with this Mail app issue can be found in the Apple Support article: If Mail on your Mac keeps asking for your password - Apple Support
"Check your password in Internet Accounts preferences
Depending on the type of email account you're using, you might be able to correct the password as follows:
- Quit Mail. If it doesn't quit, you can force it to quit.
- Choose Apple menu > System Settings (or System Preferences), then click Internet Accounts.
- Click your mail account.
- Click the Details button, if present.
- If you see a password field for your account, delete the password and type the correct password.
- Close the settings window, then open Mail and try using your account again.
If Mail still asks for your password, the password is probably incorrect. Your email provider can help you verify your email settings and make sure that you're using the correct password."
If you continue to have this trouble, get in touch with Apple Support for further assistance:
Apple Support
Best regards!