Well I assume you are running Windows XP...
Most of the time, the black screens are due to video drivers problems or sometimes old functions no longer supported.
What video card have you got?
You should check you have the latest drivers. A little visit on Windows Update shouldn't hurt too.
In the setup of Diablo II, disable Direct 3D acceleration. Anyway you will see VERY little improvement. You could also try to lower the acceleration level of your video card.
For that right-click on an empty location of your desktop and chose properties. Then go to settings and chose advanced.
Go to Troubleshooting and lower the acceleration level.
See if it solves your problem!