From the Message List of an Email Campaign, you can delete a variant by clicking on the "Delete" link next to the respective email message.  Please note that this does not delete the message in its entirety, but instead you will be taken to the next page where you are given the option to delete the variant or the original message.  Below are the exact steps:
  1. Email > Email Campaigns
  2. Click Campaigns
  3. Click Manage next to the appropriate campaign
  4. Click Messages
  5. Click Delete next to the appropriate message
  6. Click Delete next to Original Message or Variant 1 and confirm selection