Add loggedout class on body using a function or JS

Presumably your theme is using the body_class function to output classes. You can add a filter to remove or add your own classes.

function wpd_logged_out_body_class( $classes ) {
    if( ! is_user_logged_in() ){
        $classes[] = 'loggedout';
    }
    return $classes;
}
add_filter( 'body_class', 'wpd_logged_out_body_class' );