Ryan Craine hates carrying cash and finds writing checks to be a headache. He doesn’t do much of either anymore — he mostly uses his smartphone to pay for things.

Mr. Craine, a 28-year-old tech support worker in Washington, D.C., uses Apple Pay at the stores and restaurants that accept it. About 20 times a month, he turns to Venmo, a digital wallet for transferring money from one person to another, to pay his share of rent, meals, groceries and utility bills. To refinance his student loans last year, he went to an online lending start-up, Earnest.

Mr. Craine’s money choices point to the millennial-led shift toward new digital financial services, a change in behavior that threatens to upend the consumer banking industry. The popularity of the services has left the major banks rushing to adapt, even as they have regained their footing after the financial crisis.

If the banks fail to meet the challenge, Brian Moynihan, the chief executive of Bank of America, warned in November, “it may allow part of our industry to be forever taken away from us.”