Q: All of a sudden Mac Mail is asking for password

I have seen similar problems.  In my case on my Mac the accounts hat have been asking for passwords in the last few weeks have all been from the mail server formerly known as Verizon.  I assumed the issue had to do with Verizon buying all and switching many of its email customers, including me, to all servers.  Other email accounts I have on my Mac, including iCloud, Gmail and Yahoo, have seemed to work fine while the former Verizon accounts repeatedly ask for passwords.  When I paste in the correct password sometimes it is accepted, but often not.  The good news is when I ignore the problem it seems to love itself pretty soon. 

On my iPhone Mail has long had ongoing password problems in various accounts.  My solution is to ignore Mail on my phone.  The screen is too small for me to read the email anyway. 

Delete the password from Keychain Access (Applications/Utilities).

Keychain Issues - Resolve  see post by Kappy