Small. Fast. Reliable.
Choose any three.
Note: If you came here looking for a commercial solution to encrypt SQLite Databases, head over to the official SQLite professional Support page.

As SQLite databases are not encrypted or password-protected by default, ways of maintaining their security should be found. When an SQLite database is stored on the file system of your server, if it's stored in a location that is "web accessible" (whether this URL is announced or not) it is possible that this could be found, downloaded, and exploited. If you don't keep sensitive information in an SQLite database, and don't mind that others see the stucture of your backend databases, this might not be an issue for you. However, in most cases, it is.
This isn't an SQLite-specific problem, as this problem exists in other different important files, exposed to a remote user (.inc, .class, .conf)...
Not that the security of SQLite needs to be enhanced, rather, implementers of SQLite databases should be aware of the security implications and use common sense.

      $db = $_SERVER['DOCUMENT_ROOT']."/../database.sdb";
      $handle = sqlite_open($db) or die("Could not open database");

      <FilesMatch "\.(sqlite|sdb|s3db)$">
      Deny from all