How to Find a Gallery’s ID

The shortcode supports specifying the gallery’s id by supplying the id parameter. More on this in the documentation.

However, using id might be contra productive if you wish to use two different styles of galleries across your site, in which case you would need to use the same style for a whole lot of #ids in your CSS – which you might not even know beforehand. Unfortunately there is not class attribute available. To accomodate this need you could remove the default gallery shortcode and add your own shortcode handler function replicating the functionality of gallery_shortcode while adding support for a class parameter or what you might need.

remove_shortcode('gallery', 'gallery_shortcode');
add_shortcode('gallery', 'my_gallery_shortcode');

function my_gallery_shortcode($attr) {
    ...
}

EDIT: Another, perhaps simpler, option to accomplish the same thing would be the create a wrapper shorttag for gallery:

add_shortcode('my_gallery', 'my_gallery_shortcode');

function my_gallery_shortcode($attr) {
    $attr = shortcode_atts(array('class' => 'gallery-wrapper'), $attr);
    $class = $attr['class'];
    unset($attr['class']);

    return '<div class="' . $class . '">' . gallery_shortcode($attr) . '</div>';
}