Starting in version 5.8, a new backup option for text-only is available to all users (Basic, Plus, and Premium). This setting is enabled by default for Basic users. Plus and Premium users will need to enable manually, if desired. These backups are encrypted on iCloud. Please note that these backups are contingent on the available space on the iCloud account.
Please go to Day One > Settings > Import/Export > Import Text-Only Backups.
When importing one of these backups, media is not restored. These are text-only.