Hi Martin,
I’ve tried to deactivate all the plugins except Cornerstone, then the customizer is working fine again:
http://i.imgur.com/EoYI3n5.png
Please try testing for a plugin conflict. You can do this by deactivating all third party plugins, and seeing if the problem remains. If it’s fixed, you’ll know a plugin caused the problem, and you can narrow down which one by reactivating them one at a time.
Hope it helps.