If the user has a browser that does not support it, the browser will just display a regular text input box, into which the user could enter anything they like.
You'd therefore need to be able to cope with any arbitrary data being input. But that would apply anyway, since you should never trust user input coming from the browser.
If you want all users to have a nice date control, you could use a jQuery datepicker or similar to backfill the feature for browsers that don't have it.
As for what to do with the yyyy-mm-dd value when you get it -- just use the DateTime() class to turn it into a PHP DateTime object. Easy:
$dateObj = new DateTime($_GET['mydate']); //or whatever your input field is called.