One of the following steps should help.
1. Install Adobe Flash Player.
2. Enable Plug-ins
Safari > Preferences > Security
Click in the box for "Enable Plug-ins".
3. Check whether you are currently in YouTube HTML 5 trial.
At the bottom left of the page uncheck the box for
"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".
5. Turn off Extensions/Plug-ins that may interfere with Flash working properly.
Safari > Preferences > Extensions
Now try YouTube.