After Effects CC (12.0.1, 12.1, and 12.2) updates fix crash on Mac OS X v10.9 (Mavericks)
UPDATE: There is no need to use the After Effects CC (12.0.1) update any longer. Instead, install the new, full After Effects CC (12.2.1) update, which also adds Mavericks compatibility.
For After Effects CS6, install the After Effects CS6 (11.0.4) update.
Be sure to read the instructions on the update page about installing the update manually if necessary.