XF 1.2 Users can't log in.

In the log? I've not really taken a look.
Well, the first step towards getting help is giving enough information - like any errors reported in your system log and if you have access to it your web server logs. Until you start to help yourself by reviewing that information for any pertinent errors and relaying that then it's going to be hard to help you.
 
Are you using CloudFlare? If so make sure that RocketLoader is turned OFF.

When looking at your site in Chrome's inspector I see 5 warnings...

Uncaught Error: Syntax error, unrecognized expression: unsupported pseudo: -webkit-autofill
 
RocketScript/RocketLoader is definitely enabled:
Code:
<script data-rocketsrc="js/jquery/jquery-1.10.2.min.js" type="text/rocketscript"></script>
Had just checked the page source myself to see if it was or not and saw that you had already posted. That's why I specifically had asked @Thomasb454 about it since it is a BETA product and KNOWN to cause problems.:)
 
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