Juicebox-Pro does not use a key or serial number (although downloads are restricted to licensed users only).

If you still have a copy of the Juicebox-Pro zip file, you can just transfer it across to your new computer on a USB flash drive.
You can then unzip the file and install JuiceboxBuilder-Pro on your new computer following the installation instructions here.

If you need a new copy of the Juicebox-Pro zip file, you can download it again using the link from your purchase email. (The download link always points towards the latest version rather than the version you purchased.)
If you no longer have access to your purchase email, then please fill in this Download Link Request Form to the best of your ability and the admin team will send you a new download link.

Incidentally, you can install JuiceboxBuilder-Pro on up to 3 different computers, as noted in this FAQ:
Can I install on multiple computers?