Call Fargo tech support with your serial number (or license key). They will enable it to activate again.
Install and activate the software per usual on your new PC.
Copy your database file (AsureID.mdb) from your old computer to the new one.
In Windows XP, it is located here:
C:\documents and settings\all users\application data\hid global\asureid\asureid.mdb
In Windows 7, it is located here:
It is not necessary to copy over any other files from these directories; doing so will cause problems.