I used this solution to fix it on OS X 10.10:
Make sure Steam is not running and uncheck it from the accessibility preference pane.
Close system preferences.
Open Terminal and copy/paste the following exactly as you see it, including all quotes:
sudo sqlite3 /Library/Application\ Support/com.apple.TCC/Tcc.db 'delete from access where client like "%Steam%"'
Re-open the accessibility preference pane and make sure that Steam is not listed.
Close system preferences, launch Steam, and try to play a game. The warning dialog box about the Steam overlay will pop up again. Use it to go back into system preferences and Steam should have re-appeared. Check the box if it's not already, close system preferences, and you should be good to go.
Nothing extracted yet.
osx 10.9
After starting the steam and starting a game Steam client asks for this permission for steam even though the permission is already given. So every time I just have to go back to that screen and close it and then click "ok" on steam pop-up to start playing.