Ok so apparently I have been having this question & issues for months. I got so tired of pop up for pw screen. So i did my research with yahoo....
Apparently if you have a FREE yahoo account (email) , well then your SOL.... Basically.... You now have to upgrade to Yahoo Mail Plus ( http://mailplus.mail.yahoo.com/ ) the cost shows $19.99 p/yr. and in reading ABOUT Yahoo Mail Plus it mentions, the ability to use email clients such as outlook ,etc..
A BIG DISAPPOINTMENT for me, but THANK GOD I finally decided to seek help and now I got my answer. Hope this helps other frustrated YAHOO'ers that were having the same issues with Mac Mail on OS Mountain Lion for Macpro
Running Mail in Mountain lion. About every 10 mins. I am asked for my yahoo password. Have gmail account with no problems. No problems with the mail accounts on iPhone or iPad. Apple support says it's a problem with Yahoo servers. No answer from Yahoo yet. Any thoughts?
I have the same problem. Not sure exactly what the problem is, but for what I've been reading, one possibility is that Yahoo might be limiting the number of connections from your IP address. We've started having problems at my house since I've created email addresses for my kids...
The other possibility might be passwords. There is a long thread here about this (http://forums.macrumors.com/archive/index.php/t-1358234.html) and some people seem to have fixed it by changing their password. It might be worth the try!
Anyway, I still get the password window about 2-3 times a week. I hope they'll fix this because I am paying for the Yahoo Mail Plus account!