The security model for Flash Player in Safari version 10 has been changed requiring users to explicitly enable the extension and adjust the default settings before use. Below are the steps to change the settings required to enable Flash support.
To download the newest version of Flash Player, please visit https://get.adobe.com/flashplayer/.
Within Safari, open Preferences and navigate to the Security tab. Click the Plug-in Settings button.
Click the check box for Adobe Flash and change the When visiting other websites setting to Yes.
Exit Preferences and navigate to the Secured Media Library (http://go.osu.edu/SecuredMediaLibrary) and click a play link for a video. You should receive a Click to use Flash dialog in the web page's player. Click Click to use Flash.
Safari will present you with a confirmation dialog box. Click Trust.
Your video should now play properly for any videos in the Secured Media Library.
You can repeat these steps, navigating to the web site of your choice, if you experience this problem with any other site utilizing Flash similarly.