Q: After installing Maverick Mail keeps asking for passwords for all accounts.  No matter how many times I put them in I still can't get any emails

Open Keychain Access and see if you can unlock "Local Items" keychain.

Ross ~ Welcome to the Support Communities. This article may help:

Is Mail not saving your iCloud password after the Mavericks update? Here's a work-around

Ok that's the problem then.  If you have "Login" keychain present try to lock unlock it this will prompt you to Reset Password for "Local Items" keychain.  If you don't have the password or can't unlock "Login" keychain then reset default keychain from preferences and try the steps after.

In the connection doctor - if you look up the details, do you see something like "no authenticate" or "invalid userid/password"?

Essentially if you can't unlock the "Local Items" keychain then no passwords are being stored/accessed therefore it won't authenticate you.

Well, local items are unlocked, passwords are stored - but I still cannot connect to my IMAP Account. POP and SMTP are just working fine with the same account. No idea why this is happening, thought of some problems with port 143 or 993... but that is just a guess.

Since its an IMAP account have you tried to re-add it?

Yes very often. This is what I did:

I have tried different/alternative server configurations, different ports, Firewall settings, SSL on/off, imported settings via TimeMachine backup, have set a VM with a upgraded Mavericks and one clean install Mavericks - just to test the Mail settings over and over again as I really DO need it to work.

Strange thing is that everything worked perfectly fine until Mavericks and still works on ML. And now your suggestion with the Keychain, deleting the entrys, re-entering them and so one. Added Accounts via Mail and via System preferences.. nothing works.

Only thing that 'worked' was adding the same account as POP (ssl & port 110). On every other device (iPhone, iPad, MBP2012) everything works perfect.

Update:

Every IMAP setting gives me this:

Thank you, Alancito.  I was having the same problem.  The workaround fixed it.

Big help...Thanks!

Worked for me..... Thank you Alancito!