Example of how you can include the event ID's for the registrations you currently have in your session within the checkout page body classes.

functions.php

<?php //Please do not include the opening PHP tag if you already have one

function tw_ee_return_event_id_on_spco_body_class( $classes ){
    // get out if this isn't the reg checkout page
    if ( ! class_exists('EE_Registry') || ! is_page( EE_Registry::instance()->CFG->core->reg_page_id ) ){
        return $classes;
    }

    $events = array();
    $checkout = EE_Registry::instance()->SSN->checkout();
    if ( $checkout instanceof EE_Checkout ) {
        $transaction = $checkout->transaction;
        if ( $transaction instanceof EE_Transaction ) {
            foreach ( $transaction->registrations() as $registration ) {
                if ( $registration instanceof EE_Registration ) {
                    $event = $registration->event();
                    if ( $event instanceof EE_Event ) {
                        $events[] = 'event-id-' . $event->ID();
                    }
                }
            }   
        }
    }
    $events = array_unique($events);
    return array_merge( $classes, $events );
}
add_filter( 'body_class', 'tw_ee_return_event_id_on_spco_body_class' );