Add a class to the body tag when a menu is enabled

Use the has_nav_menu to detect if the menu is configure in a body_class filter.

Something like

add_filter( 'body_class', 'wpse209469_body_class' );
function wpse209469_body_class( $classes ) {

    if (has_nav_menu('primary-menu'))
      $classes[] = 'primary-menu';

    return $classes;
}

If you have two menus you can change the condition to match both, or just do two filters. There is probably a way to enumerate all the menus and add classes for them but it is probably an overkill.