Determine if a navigation item has children

You can filter wp_nav_menu_objects and add the classes in one rush. You don’t even need a custom walker for that.

add_filter( 'wp_nav_menu_objects', 'add_has_children_to_nav_items' );

function add_has_children_to_nav_items( $items )
{
    $parents = wp_list_pluck( $items, 'menu_item_parent');

    foreach ( $items as $item )
        in_array( $item->ID, $parents ) && $item->classes[] = 'has-children';

    return $items;
}

The class has-children will now be assigned to the matching li elements automatically.

In a custom walker you will find the class in $item->classes in your function start_el(). The following lines from the default walker will do the trick:

$class_names = join( ' ', apply_filters( 'nav_menu_css_class', array_filter( $classes ), $item, $args ) );
$class_names = $class_names ? ' class="' . esc_attr( $class_names ) . '"' : '';