The general concept on dynamic sidebars

Basically WP loops through the global $wp_registered_widgets and builds the widget depending on it’s arguments. You can intercept them easily:

Use a filter

You can use the dynamic_sidebar_params to add a class to target specific widgets:

// SHOW the params for better insights
function wpse44903_dump_sidebar_params( $params )
{
    echo '<pre>';
    print_r( $params );
    echo '</pre>';

    return $params;
}
add_filter( 'dynamic_sidebar_params', 'wpse44903_dump_sidebar_params' );

Then you can simply alter whatever you need (depending on your situation):

// ALTER the params to your needs
function wpse44903_alter_sidebar_params( $params )
{
    // remove trailing and leading white space and use only lower case characters to be on the save side of things
    if ( 'YOUR WIDGET NAME' === trim( strtolower( $params[0]['widget_name'] ) ) )
    {
        // do whatever you need to do.
        // @example:
        foreach ( $wp_registered_widgets as $widget )
            if ( in_array( $params[0]['widget_name'], $widget ) )
                print_r( $widget );
    }

    return $params;
}
add_filter( 'dynamic_sidebar_params', 'wpse44903_alter_sidebar_params' );

Note: This is not tested and maybe needs some bug fixing. Be sure to turn WP_DEBUG to TRUE in your wp-config.php file.