jQuery UI Datepicker not working

I often type things wrong. So, I would start debugging you copying and pasting the links to the JS scripts in your browser and make sure they load.

Then in Chrome go to the Wrench Menu -> Tools -> JavaScript Console. Here you will be able to type/execute your JavaScript directly. I would start off my typing jQuery into the console to make sure jQuery is actually loaded. Then try doing jQuery('#datepicker') if it returns empty brackets [] then your selector is failing. If it works, try opening up the datepicker – you will probably see an error in the JS console.

Like others have suggested, I think the problem is that the script is running before the is actually rendered. I would suggest doing the following:

<script type="text/javascript">
jQuery(document).ready(function() {
    jQuery('#datepicker').datepicker({
        dateFormat : 'yy-mm-dd'
    });
});
</script>

This will force the script to run after the entire page has loaded.

== EXAMPLE ==

add_action( 'init', 'wp29r01_date_picker' );
function wp29r01_date_picker() {
    wp_enqueue_script( 'jquery' );
    wp_enqueue_script( 'jquery-ui-core' );
    wp_enqueue_script( 'jquery-datepicker', 'http://jquery-ui.googlecode.com/svn/trunk/ui/jquery.ui.datepicker.js', array('jquery', 'jquery-ui-core' ) );
}

add_action( 'wp_footer', 'wp29r01_print_scripts');
function wp29r01_print_scripts() {
    ?>
<script type="text/javascript">
    jQuery(document).ready(function() {
        jQuery('#datepicker').datepicker();
    })
</script>
    <?php
}