I want to export private images from Drupal to an external client. How to I authenticate an external client to access private files from Drupal DB?
You can use file_create_url($uri) to create an externally accessible url for the file.
If used with private file mode files, you will need to implement hook_file_download
You might also be interested in the Services module, to expose an API for the files, which can provide authentication methods such as Drupal session and OAuth authentication.