Apple Mail keeps asking for google login since Sierra 10.12.4 Beta
I am not sure what else you've tried so far, so I'll begin from scratch:

Disable two-step verification

In the past, I've found that two-step verification caused issues between OS updates. Use Google's website to disable two-step verification for your accounts. You can reenable it later. If you aren't using two-step verification, continue.

Re-authenticate


Completely remove the affected accounts: System Preferences > Internet Accounts > select account > click minus icon
Add the accounts again. You should see a Google-branded login frame in the window.


If you are able to do this, continue monitoring for the issue. If you've already tried this, please update your answer with what is currently happening and the steps you've taken.