There is a menu option under Help to "Recover Key". Type in the email address you originally used to register and a copy of your key will be sent.
Download and install a new copy of Beersmith on your new computer.
Make sure the version numbers are the same on each computer.
Make sure that beersmith isn't running on either computer during the following step.
Copy the *.bsmx and *.bxml files from the folder where beersmith is installed on the old computer to the same location on the new computer. It is probably at c:\program files\beersmith2 if you are on windows.
If that doesn't work, try the beersmith forum http://www.beersmith.com/forum/index.php