If you checked that you have wp_head and wp_footer in your themes header.php and footer.php you can find additional suggestions for getting it working here: http://contactform7.com/why-isnt-my-ajax-contact-form-working-correctly/
Either way, the ajax submit looks more fancy and is always good to use (clearly gives a better user experience) but your problem is CSS related anyway. So you might want to fix your CSS for the users with disabled JS anyway.
You just have to override the .wpcf7-not-valid class with your own custom css styles.
Additionally your #footer overlaps with the submit button of the form. I was not able to submit the form until I removed the footer div from the DOM (Firefox 20 under Ubuntu).