Yes, I've done it.
I'm not sure whether you need to use Selenium, but I would recommend it. Stripe uses javascript, so you would need a javascript-capable tool to fully test the interaction. Selenium is a good option. I use Codeception with Selenium Webdriver for these tests.
I found one gotcha. Because I used Stripe's jquery.payment library to enhance my form, the form input is reformatted as you type it (e.g. type the whole card number, and jquery.payment automatically puts the correct spaces in).
If you use this too, be aware that this javascript text formatting causes problems with Webdriver: it sometimes fails to enter all the text for a field. To get around this, you can insert a delay like so:
// Slowing this down so that javascript formatting doesn't break
$I->wait(0.2);
$I->appendField('Card number', '3714');
$I->wait(0.2);
$I->appendField('Card number', '496353');
$I->wait(0.2);
$I->appendField('Card number', '98431');
$I->wait(0.2);
It's a bit ugly and hacky, but it works. The text must be split in the same way as it will be formatted; in this example, it's an American Express card.