Safari 7.0 and 6.1 include a new feature called Safari Power Saver, which is intended to disable plug-ins in order to conserve battery power. 

When hovering over the video window of a presentation, there should be an icon that appears which you can click on in order to allow the plug-in on that webpage.  However, we have seen cases where this icon does not appear properly.  If this happens, you must disable Power Saver completely in order to allow the plug-in to run.

To disable the Power Saver function, enter Safari Preferences

Then go to the Advanced tab, and uncheck the box to "Stop plug-ins to save power". 

Reload the webpage containing the Mediasite presentation and it should play properly.