Does at&t yahoo web mail support browser Safari Version 12?

I don't know what that means.  Who's support?  Is safari inadequate of displaying and handling web pages that all other browsers can display or is the att webmail inadequate of supporting the latest version of safari running on Mojave and now needs to change to cater what the new safari requires?  You said it yourself.  You can run chrome under Mojave and display the webmail with no problems but you then run safari on that same OS and you have problems.  So wouldn't you think the problem points to safari or at least safari running on Mojave as the one having the problem?  So I would say it's an apple support problem.  Most probably a bug.

Maybe you should wait until the "new" webmail interface comes out and use chrome in the meantime.  Maybe you will "get lucky" but I doubt it.

By the way, do you have any extensions installed in your safari?   If so try disabling them.

FWIW both the webmail account and settings buttons display drop down menus while home is just brings up another web page.  So I'm speculating that safari (or the OS preferences) have some kind of settings of its own controlling drop down/popup menus on web pages (why? I can't guess) or safari just doesn't handle drop down/popup menus (maybe a bug).  Just out of curiosity go to the att web/portal page, start.att.net.  Assuming you are already logged in click the account box just to the left of the mail icon in the upper right.  Does that work to bring up its drop down menu?