How to make the Newsletter plugin visible to users with author privileges?

The plugin contains for example these lines:

add_menu_page('Newsletter', 'Newsletter', 
              ($this->options['editor'] == 1) ? 'manage_categories' : 'manage_options', 
              'newsletter_main_index');

add_submenu_page('newsletter_main_index', $title, $title, 
              ($newsletter->options['editor'] == 1) ? 'manage_categories' : 'manage_options', 
              $name, $name);


add_submenu_page(null, $title, $title, 
              ($newsletter->options['editor'] == 1) ? 'manage_categories' : 'manage_options', 
              $name, $name);

so it looks like you have the option to display the Newsletter menu for editors (manage_categories) or admins (manage_options).

If this menu is going to be accessible for authors, you would need for example the publish_post capability.

Instead of modifying the plugin directly, and loose the modification on each update, you should try to ask the plugin author to add this as an option.

In the meanwhile you could try to add the manage_categories capability to the authors, if you are happy with that kind of temporary quick fix.