Had to change gmail password on 2 accounts and add 2 step verification. One works fine, the other will not download new messages. No error message or prompt for password. Turned 2 step verification off on that one, no difference. Turned back on. Very frustrating!
Please check if eM Client is on the list of allowed apps in your Google account security settings on the problematic account. If it is, please try removing eM Client from the list, then removing and re-adding the account in eM Client. It will prompt you to allow access again and should work the second time around.