From the FAQ page on the BeerSmith.com web site:
How can I reinstall my copy of BeerSmith 3 on a new computer?
First make a complete copy of your Documents/BeerSmith3 directory and save it to a removable drive or thumb drive
Next download, install and BeerSmith 3 on your new computer. Run it once and then shut it down.
Finally make sure BeerSmith 3 is shut down, and then copy the entire BeerSmith3 directory from the old computer (via thumb drive or removable drive) to the Documents/BeerSmith3 directory on the new computer. Overwrite any data there.
Activate the new copy of BeerSmith on your new computer. You may need to go to your manage licenses page to deactivate your old computer before you can activate the new one.