PSA:
I'm playing D2 on a Mac, I'm still using High Sierra 10.13.6, and I have a ton of trouble getting the game to start. I usually have to click on the icon, wait for it to bounce a little, then force quit, and start the process all over again. This can take days before I finally get lucky and get the game to actually start up. Recently, I decided to try reinstalling D2 and D2LoD and, amazingly, all issues starting up appear to have been fixed!
1. Install the latest security update and go through the restart
2. Delete your current D2 and D2LoD files (make sure to back up any of your character files!)
3. Download the proper installers from Blizzard and go through the installation walk-through
4. Go to battle.net to install the latest patch, it might look like it's not working once the "Patching..." bar hits 100% and starts to hang, but be patient, it will give you a success message eventually.
5. You're done, play and be merry
I have no idea if this works in Mojave, but if you ever have issues getting the game to start, try re-downloading and installing fresh copies from Blizzard.