After Effects CS6 should run OK in OS X 10.9.3.
Is there anything that makes you think it won't?
FOR MAC USERS
-Mac 10.9.3 workaround https://forums.adobe.com/thread/1489922
-more Mac 10.9.3 https://forums.adobe.com/thread/1491469
-Enable Mac Root User https://forums.adobe.com/thread/1156604
-more Root User http://forums.adobe.com/thread/879931
-and more root user http://forums.adobe.com/thread/940869?tstart=0
-Mac 10.9.3 and CS6 https://forums.adobe.com/thread/1480238