Go step by step and test.
1. System Preferences > Flash Player > Advanced > Delete All
Press the "Delete All" button
Install Adobe Flash Player.
http://get.adobe.com/flashplayer/
Follow the prompts. Click Safari in the menu bar and then Quit Safari .
Restart computer. Relaunch Safari.
2. Allow Plug-ins
Safari > Preferences > Security
Internet Plug-ins > "Allow plug-ins"
Enable it.
Click “Done”.
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. Delete cookies.
Safari > Preferences > Privacy > Cookies and other website data:
Press the “Remove All Website Data” button. It may take a while.
5. Turn off Extensions, if any.
Safari > Preferences > Extensions
Relaunch Safari.
Now try YouTube.