If you override the view mod/file/views/default/resources/file/view.php with a modified version where you enclose the registered of the download title menu button with

if (elgg_is_logged_in()) {
    elgg_register_menu_item('title', array(
        'name' => 'download',
        'text' => elgg_echo('download'),
        'href' => elgg_get_download_url($file),
        'link_class' => 'elgg-button elgg-button-action',
    ));
}

only users who are logged-in to your site will see the download button whereas logged-out visitors will only see the file (as long as its access level is public) but don't see the download button.

Learn how to override views: http://learn.elgg.org/en/stable/guides/views.html#overriding-views. That's better than modifying the file plugin file in the long run.

Thanks for pointers.

I am working on Elgg project (A Community Site of my own). This helped me get my hands dirty with Elgg.

I have converted the requirement to Elgg plugin. So people can install the plugin instead of changing Core file. Below is plugin link

https://elgg.org/plugins/2958220