It is not an either or proposition, you can use both ids and classes on elements. Generally, it is a good idea to do both. Drupal code by default does on many things.
In your case for instance you would want something like <div id="menu-1" class="menu-bar", <div id="menu-2" class="menu-bar".
Then you can do .menu-bar for general styles and #menu-1 for styles specific to that menu.
Hope that is helpful, ~~nat