1. System Preferences > Flash Player > Advanced > Delete All
Press the "Delete All" button under “Browsing Data and Settings”.
Install Adobe Flash Player.
http://get.adobe.com/flashplayer/
Download it first.
The next step is important.
Click Safari in the menubar and select “Quit Safari”.
Follow the prompts and install it.
Restart computer. Relaunch Safari.
2. Allow Plug-ins
Safari > Preferences > Security
Internet Plug-ins > "Allow plug-ins"
Enable it.
3. Safari > Preferences > Privacy > Cookies and website data:
Press the “ Remove All Website Data” button.
Press the “Details” button.
Remove YouTube related cookies.
4. Safari > Preferences > Extensions
Turn all extensions off, relaunch Safari and test.
Turn those on one by one and test.