Use a plugin for that. Hook into `after_setup_theme` and check if the current theme is the correct child theme:

    add_action( 'after_setup_theme', 'common_child_theme_functions' );
    
    function common_child_theme_functions()
    {
    	if ( ! 'Twenty Twelve' === wp_get_theme()->parent() )
    		return;
    
    	// do your work
    }

[`wp_get_theme()`][1] returns a `WP_Theme` instance, and `parent()` either the parent theme name or `FALSE`.

I do not recommend a mu-plugin. Real plugins can be deactivated, and you can enable or disable them for each site separately. Adjust the capabilities for the clients instead; this is a much cleaner approach.


  [1]: http://queryposts.com/function/wp_get_theme/