Form has a div within it. How can I add classes to this div?
That's not what you want. What you want to do is add it to the form structure like this:

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


Since you didn't list an element on the form, I can only assume you want the class on the form itself. If you want to wrap an element with a div with a class on it, you can do:

$form['element']['#prefix'] = '&lt;div class="my-class"&gt;';
$form['element']['#suffix'] = '&lt;/div&gt;';


That will wrap the entire element.