Wordpress - Add 'has_children' class to parent li when modifying Walker_Nav_Menu

start_el() should get this information in its $args parameter, but it appears WordPress only fills this in if $args is an array, while for the custom navigation menus it is an object. This is reported in a Trac ticket. But no problem, you can fill this in yourself, if you also override the display_element() method in your custom walker (because this is the easiest place to access the child element array):

class WPSE16818_Walker extends Walker_Nav_Menu
{
    function display_element( $element, &$children_elements, $max_depth, $depth=0, $args, &$output )
    {
        $id_field = $this->db_fields['id'];
        if ( is_object( $args[0] ) ) {
            $args[0]->has_children = ! empty( $children_elements[$element->$id_field] );
        }
        return parent::display_element( $element, $children_elements, $max_depth, $depth, $args, $output );
    }

    function start_el( &$output, $item, $depth, $args ) {
        if ( $args->has_children ) {
            // ...
        }
    }

Update: As of WordPress 3.7 (October 2013), CSS classes have been added to indicate child menu items and pages in theme menus — no need to use a custom walker as it's taken care of in WordPress core.

The CSS classes are named menu-item-has-children and page_item_has_children.

---

For a complete solution for anybody in a hurry (credit to Jan Fabry's previous answer), see the full implementation below.

Output the navigation in your theme's template:

wp_nav_menu( array(
    'theme_location' => 'navigation-primary',
    'container' => false,
    'container_class' => '',
    'container_id' => '',
    'menu_class' => '',
    'menu_id' => '',
    'walker' => new Selective_Walker(),
    'depth' => 2
    )
);

Then, include the following in your theme's functions.php:

class Selective_Walker extends Walker_Nav_Menu {
    function display_element( $element, &$children_elements, $max_depth, $depth=0, $args, &$output ) {
        $id_field = $this->db_fields['id'];

        if ( is_object( $args[0] ) ) {
            $args[0]->has_children = !empty( $children_elements[$element->$id_field] );
        }

        return parent::display_element( $element, $children_elements, $max_depth, $depth, $args, $output );
    }

    function start_el( &$output, $item, $depth, $args ) {
        if ( $args->has_children ) {
            $item->classes[] = 'has_children';
        }

        parent::start_el(&$output, $item, $depth, $args);
    }
}

The resulting HTML output will resemble the following:

<ul>
    <li><a href="#">Home</a></li>
    <li class="has_children"><a href="#">About</a>
        <ul class="sub-menu">
            <li><a href="#">Our Mission</a></li>
        </ul>
    </li>
    <li><a href="#">Services</a></li>
    <li class="has_children"><a href="#">Products</a>
        <ul class="sub-menu">
            <li><a href="#">Lorem Ipsum</a></li>
            <li><a href="#">Lorem Ipsum</a></li>                
        </ul>
    </li>
    <li><a href="#">Contact Us</a></li>
</ul>

For more information on using WordPress' walker class, see Understanding the Walker Class.

Enjoy!

This function is exactly what you want to have. It also shows you a pretty effective way to modify nav menu items. Furthermore you can open it for more advanced (eg. child theme) functions via the item-filter:

/**
 * Classes for a navigation named "Topnav" in the nav location "top".
 * Shows examples on how to modify the current nav menu item
 * 
 * @param (object) $items
 * @param (object) $menu
 * @param (object) $args
 */
function wpse16818_nav_menu_items( $items, $menu, $args )
{
    # >>>> start editing

    // examples for possible targets
    $target['name'] = 'Topnav';
    // The targeted menu item/s
    $target['items'] = array( (int) 6 );

    # <<<< stop editing

    // filter for child themes: "config_nav_menu_topnav"
    $target = apply_filters( 'config_nav_menu_'.strtolower( $target['name'] ), $target );

    // Abort if we're not with the named menu
    if ( $menu->name !== $target['name'] ) 
        return;

    foreach ( $items as $item )
    {
        // Check what $item contains
        echo '<pre>'; print_r($item); echo '</pre>';

        // First real world example:
        $item->classes = 'span-4';

        // Second real world example:
        // Append this class if we are in one of the targeted items
        if ( in_array( (int) $item->menu_order, $target['items'] ) )
            $item->classes .= ' last';
    }

    return $items;
}
add_filter( 'wp_get_nav_menu_items', 'wpse16818_nav_menu_items', 10, 3 );

And yes, there's - in nearly every case - no need for a custom walker.