If the software is already on your machine:

On Windows 7, go to: Start -> All Programs -> GenArts SapphireAE -> Activate Sapphire AE License.
On Windows 8, search for GenArts SapphireAE -> Activate Sapphire AE License.
On Mac, go to: /Applications/GenArtsSapphireAE folder and double click on "Install Sapphire".

• Choose to "Activate your license manually using another computer's Web browser".
• Get the key file from your computer that has Internet connection and transfer it to your offline machine via a flash/thumb drive, etc.
• Select the location of the key file, and pick a location for the request file (which will be created).
• Copy the request file (sapphire.req) to a machine with Internet.
• Upload it to http://activation.genarts.com/offline-activation.php.
• Save the activation file it returns (via download or email), and copy that back to the offline machine.
• Enter its location into to the license tool.

For floating license instructions, go here.