If you click the "Browse" button and a dialog window doesn't appear, you probably have Flash Player 10 installed. Adobe changed a number of security-related things in player 10, and as a result broke file uploading both in our app as well as many other sites that use Flash to facilitate file uploading. We have now fixed these issues in Director 1.2.9, so upgrade to the latest release to bring Flash 10 compatibility to your Director install.
To see if you have Flash 10 installed, visit this test page.
Click here to learn how to upgrade.
If upgrading is not an option for you, the only other way to fix this would be to downgrade to Flash Player 9. You can get old versions of the Flash Player here.