You mean Square. A little of both. The WSJ reported that their gross margin shrank from 27% to 21% in 2013 and specifically mentioned fraud as a component. (Interchange didn't change much.) [1]
With very easy signup with minimal due diligence, plus next day payout, it's an attractive target for fraudulent sellers. Even with relatively honest but low volume sellers, chargebacks may be hard to collect once the money's paid out, so Square is left holding the bag. With a gross margin of barely more than half a percent of the transaction amount, one bad merchant can erase the revenue from a great many good ones.
Stripe and PayPal face similar issues in the ecommerce world, and you can see how annoying PayPal has had to become to tighten up their fraud controls (lots of bad experiences with funds getting frozen).
Hmm, fraudulent sellers. So you are basically saying that if someone charges a bunch of customers and then disappears, the chargebacks fall on Square's lap? This doesn't sound right. I'd guess that this is acquiring bank's problem and they hedge against this through their fee structure.
Usually the payment processor (square in this case) is responsible for chargeback/fraud costs. The acquiring bank will only end up paying those costs if square goes under and is unable to.
Even if it's passed back to square via fees, square's still paying for it. The fact that their gross margin shrank indicates those costs increased. And we know interchange didn't account for it.
Fraud costs them money the same way support costs them money - it's an operational expense. What I don't know if they are also on a receiving end of any penalty fees, especially in cases of fraudulent merchants.
Are you making educated guesses or do you actually know that fraud costs Stripe money?
EDIT - Yep, sorry, Square. I'm just in the middle of integrating with Stripe, so it just slipped.