I'm having the same problems with both Google Chrome and Safari. I'm on a MacBook Pro, running 10.8.2. Both Safari (6.0.2) and Chrome (23.0.1271.64) are up to date.
I cannot use the left mouse button to click on any buttons or droplists. I have tried logging out of Ancestry, closing all of the Ancestry tabs, and rebooting, but it didn't make a difference.
Firefox 16.0.2 seems to work fine.
Edit: After 2 reboots, and deleting all of Safari's cached data, both Safari and Chrome are now working fine. One other data point - when it wasn't working I was logged in from a wifi in a coffee shop. Now I'm at home and it's working. Any thoughts?