I downloaded and installed the game on my 64-bit Linux Mint 19.2 laptop. When I click on the icon on either my desktop or Start Menu, nothing happens. When I go to my terminal and attempt to start it with the 'start.sh' command, I get this error:

kevin@kevin-Inspiron-1764:~/GOG Games/Blade Runner$ ./start.sh
Running Blade Runner
Starting ScummVM
/home/kevin/GOG Games/Blade Runner/game/scummvm/scummvm_x86_64: error while loading shared libraries: libSDL2_net-2.0.so.0: cannot open shared object file: No such file or directory
Done.

I downloaded and installed libSDL2_net-2.0.so.0, but the problem still occurs. Ideas?

EDIT: Operator error. I downloaded the correct version for my distro, and it works fine now...

How did you get it to work? I have downloaded the linux version of the game and ran all the correct commands and it still will not run. I get the exact same message as "Running Blade Runner
Starting ScummVM
/home/dstas/GOG Games/Blade Runner/game/scummvm/scummvm_x86_64: error while loading shared libraries: libmad.so.0: cannot open shared object file: No such file or directory
Done."

I have updated everything thing I have made sure that I have downloaded the latest version of of all my system package including installing scummvm.

For the missing libSDL2_net-2.0.so.0 error, the shell command to download and install (it will require root privileges) should be something like:
sudo apt-get install libsdl2-net-2.0-0

For the missing libmad.0.0, it should be something similar:
sudo apt-get install libmad0

And so on (it might prompt you that other libraries cannot be loaded).

However, you could alternatively try the flatpak version of ScummVM (or the snap version) that is available from the ScummVM website to download and bundles together all the required stuff to run ScummVM. In this case, after installing ScummVM, you'd need to add the game manually to the ScummVM launcher, by clicking on "Add Game..." and pointing it to the Blade Runner folder.