1. Install Adobe Flash Player.
http://get.adobe.com/flashplayer/
Quit Safari.
Restart computer. Relaunch Safari.
2. Enable Plug-ins
Safari > Preferences > Security
Web Content: > "Enable Plug-ins".
or
Internet Plug-ins > "Allow all other plug-ins
3. Check whether you are currently in YouTube HTML 5 trial.
http://www.youtube.com/html5
At the bottom left of the page uncheck the box beside
"You are currently in the HTML5 trial".
4. Reset Safari.
Click Safari in the menu bar.
From the drop down select "Reset Safari".
Uncheck the boxes next to all items.
Check the box next to "Remove all website data".
Click "Reset".
5. Turn off Extensions, if any.
Safari > Preferences > Extensions
Now try YouTube.
Best.