Edit the (automatic) HTML layout of nav menus

Question

Update: The question was actually how to wrap the submenus, not the whole menu. Here are 2 options:

using jQuery:

<script type="text/javascript">

// run this before the script that sets up the dropdowns:
jQuery(document.ready(function($){
    $('.sub-menu').wrap('<div class="dropdown"></div>');
})

</script>

using PHP 5.3+ (slightly more complex option, but thought i’d post it):

<?php
$dom = DOMDocument::loadHTML(wp_nav_menu(array(
    'menu' => 'main', // change this to your theme location
    'echo' => false,
)));

// Find all uls
$uls = $dom->getElementsByTagName('ul');

// Find & wrap each sub menu
foreach ($uls AS $ul) 
{
    if ($ul->getAttribute('class') == 'sub-menu')
    {
        // create the dropdown wrapper div
        $wrapper = $dom->createElement('div');
        $wrapper->setAttribute('class', 'dropdown');

        // Replace ul with this wrapper div
        $ul->parentNode->replaceChild($wrapper, $ul);

        // Append this ul to wrapper div
        $wrapper->appendChild($ul);
    }
}
echo $dom->saveHTML($dom->getElementsByTagName('div')->item(0));

?>

// OLD ANSWER:

Check out the container_class option of wp_nav_menu(): http://codex.wordpress.org/Function_Reference/wp_nav_menu

Your menu’s ul will automatically be wrapped in a div; you can set the class of the div using that option:

<?php
wp_nav_menu(array(
    'theme_location' => 'my_location', // change this to your theme location
    'container_class' => 'dropdown',
));
?>