Stripe.js is an API -- developers use this API to build something used by their customers. The customer is the one who's data is being collected, and the developers are the one's facilitating that collection via their service. The fact that it got sent to Stripe is not really relevant to who bears responsibility on clarifying data rights to the customer.
"The simplest way for you to be PCI compliant is to never see (or have access to) card data at all. Stripe makes this easy for you as we can do the heavy lifting to protect your customers’ card information." [1]
Interesting question whether Stripe incurs statutory privacy duties to the web vendor and the buyer separately. I would imagine so, because given the "triangular" nature of this kind of Stripe transaction, Stripe ends up collecting data from two parties.
[This is not legal advice]
They get the burden, but they wouldn't be able to know about evil activities from the third party provider.
In the unlikely case where Stripe was a bad player, the customer would sue the website, the website would countersue Stripe