I was going to simply put this in the comments, but I felt it's enough information to warrant an answer.
There are some nasty anchovies out there if you start looking for "key grabbers" or whatnot, so be careful. The one I found that actually worked was linked from here. It only works with version 1.12 (or possibly older); but you did say "old gaming rig" so hopefully it was old enough that the game was fairly old too. ;) If you patched to 1.13, use the Google and you should find something; look for "Serby's" or "D2Utilities".
Basically, most of them read the MPQ files from your game directory and show you the decrypted game-code (CD key), which you can then register with Blizzard/Bnet to download the new modern-OS-compatible game client.
You do this by logging in to battle.net, going to Account Settings, and clicking "Add a Game Key" (linked Q/A explains a bit more about why it works this way). Enter the code shown by the viewer for Diablo 2, then for LOD, and you will now have both games in your account. You then go to "Download Clients", "Classic Games", find Diablo 2 / LOD, and download the OS-appropriate client (Windows or Mac). When you install the game from these clients, you will still need the "game key", which you can find in your account under "Account Settings", "Summary" -- scroll down to the bottom under "Classic Games" and click on the game title. It'll look something like this:
11/11/2017 11:11 PM
Copy that long alphanumeric code & paste it into the requisite box during installation. The new installers will get you all the way up to the current patch level. Side-note, in modern Windows (7 & up) your saved character files will be in
\YourUserFolder\Saved Games\Diablo II\, which is kinda nice for management/backup/mule-ing. Have fun!