To share an asset with your entire institution, you'll need to navigate to the Sharing options window. Click the Media options icon for the asset you want to share, then click "Share."
The third sharing option in the drop-down menu is "Share with all Warpwire Support Account users (Institution)." This label will change based on your institution's name. If you choose this option, anyone who has an institutional account can log in and view your asset.
See also: Share with everyone on the web (Public) and Share with Users and Groups (Protected).