Using jQuery to retrieve customizer value

Use that JQuery code from your question in customizer preview for real time update. Your transport should be set to ‘postMessage’. In your website you can add body class in function.php:

function theme_prefix_body_class( $classes ) {

    if ( get_theme_mod('header_fixed', 0 ) == true  ) {

        $classes[] = 'fixed-header';
    }

    return $classes;
}
add_filter( 'body_class', 'theme_prefix_body_class' );