There doesn’t seem to be an easy way to do this. However, you can try this rather hackish approach:
add_filter('dynamic_sidebar_params', 'my_sidebar_params_cb');
function my_sidebar_params_cb($params) {
global $my_widget_counter;
if (empty($my_widget_counter)) $my_widget_counter = 1;
else $my_widget_counter++;
$params[0]['before_widget'] = str_replace('class="', 'class="widget_nr_'.$my_widget_counter.' ', $params[0]['before_widget']);
return $params;
}