If you have the Pro version, use the "Pack Data" command:
Asutype will wrap all data files and settings into the single packed file. You can backup this file to any backup medium such as an USB, a DVD, your Dropbox account etc. To restore it, simply use the "Unpack Data" command.
If you use the Standard version, backup all the Asutype's data folder normally. You can go to the data folder by right clicking a file's tab and choosing the "Open containing folder" command.