OK, having logged in as Admin, I can see the issue straight away, and the ‘edit post’ popup previously looked at – is affected when logged in as Admin, not guest.
I will just ask you to confirm as Andy asked, if you are running any plugins that only have an effect on Admins, as the ui-dialog (popups) appearance could be the result of some kind of js conflict.. chat.js maybe? They don’t appear to be suffering any CSS problem I can see…
I don’t know enough about js to be sure, maybe Andy can offer some insight, but I would definitely recommend temporarily disabling plugins until you find the cause.