I had the problem with two different outlook accounts, one had the 2-steps authentication one had not.
My client on both a desktop and a laptop was office outlook 2013.
Over the last few days, I was able to go by using an app password for the 2-steps account. Sometimes entering it once would suffice, sometimes it would keep asking but closing the error message would work.
For the regular account, entering the password once every time I would open office outlook would usually work.
Now today after MSFT supposedly fixed the issue, I ran into a new issue on the account that had the 2-steps authentication. I could get my emails to updates using my usual method, but when I would try to reply to an email, outlook would close abruptly.
I don't know if this was the result of the problem and fix or a coincidence, but I had to do an online repair (reinstall) of outlook in order to fix the issue.
I also has to check the remember password box the first time I got the prompt.
Since then, so far, it appears to be working.