The Apple Pay service makes it easy for consumers to pay for purchases with an Apple iPhone 6. But some banks are now making it tougher for customers to set up an Apple Pay account.

Banks are clamping down on their verification processes for the Apple Pay mobile-payment system after crooks started loading stolen credit-card data onto phones and then making fraudulent purchases. Some of that stolen card information has been tracked back to accounts that were compromised in Target’s big data breach at the end of 2013 and the...