How to style overall form
You can add classes you need directly to your form with #attributes. This is example directly from the official documentation:

$form['#attributes'] = array('class' =&gt; array('search-form'));


If it's not your code that creates form, you can add this with hook_form_alter(). Just be careful not to overwrite any attributes or classes that might be already in place:

$form['#attributes']['class'][] = 'search-form';