Mavericks handles accessibility on a per-application basis, so instead of enabling assistive devices (option does not even exist) carte-blanche, the app which makes the request shows up in a list in
System Preferences > Security & Privacy > Privacy where you can enable the access it's looking for.
Synergy, however, makes a check for the old style of assistive devices setting and fails to start.
Update: For those catching up with this issue - Synergy seems to have resolved this with 1.4.17 release.
Also, for those experiencing problems after upgrading, please see James Valero's input below.