Logic Pro Support gave me the following solution, for those who are experiencing this problem. A new High Sierra/Mojave feature requires certain permissions to be checked (allowed) in your mac privacy settings for various programs to access your libraries. When you open the Media Browser, Logic Pro is looking for all your media, including Photos, and therein lies the issue. The permission for Photos is apparently turned off by default, so the system will not allow Logic Pro's Media Browser to access any items in your Photos library. I was told that the first time this happens after you've upgraded to the new OS, there is a brief popup window to inform you of this, but if you miss it the first time, or it doesn't show at that point, there is no way for the user to know the source of the problem. Apple says they are working on some method of informing users of this within Logic Pro to eliminate the hanging problem. So, for now, here's the solution...
Close out of Logic Pro X. Then go to System Preferences, click on Security & Privacy, click the lock to make changes, choose the Privacy menu, scroll to Photos, place a check mark in the box next to Logic Pro X. While in the Privacy menu, be sure to check your other permissions, for Microphone, Contacts, Calendar, Automation, iTunes .app, and Finder to make sure they're checked also. In my case, all were checked, except Photos, which is apparently the one that causes the hang.
Thanks to the Logic Pro techs at Apple Support!