If you have BeerSmith 2 you need to make a full backup of your Documents/BeerSmith2 directory which contains all of your data by default. Your recipes are in the file Recipe.bsmx under that directory but grab a copy of the whole thing to be safe.
You can use the same registration code on the new computer (check Help->Activate on the old one if you don't know what it is) and download/install the latest version from our website.
Brad