jQuery UI Tabs - Automatic Height
All you have to do is to set a min-height. (It's taken me ages to find the answer to this .. I hope it helps!).
Here is my code that really works:
$("#tabs").tabs().css({ 'min-height': '400px', 'overflow': 'auto'});
In jQuery 1.9, use the heightStyle attribute (docs here)
$( ".selector" ).tabs({ heightStyle: "auto" });
After reading over the documentation, it seems that option is no longer available. However, it is very easy to replicate this functionality.
Assuming your tabs are arranged horizontally:
$("ul.tabs a").css('height', $("ul.tabs").height());