Drupal - How can I prevent text areas from being resizable?

It's JS that makes the text areas resizable, not CSS. I think the simplest way to disable this behaviour for all textareas is to implement THEMENAME_textarea() in your theme and remove it there:

function THEMENAME_textarea($element) {
  $element['element']['#resizable'] = false ;
  return theme_textarea($element) ;
}

If you only need CSS3 compatibility, you can use the following CSS:

textarea { resize: none; }

Or if you need to disable resize on a specific textarea, add the ID of that text area:

.edit-body-und-0-value { resize:none }