Outgoing New York Superintendent of Financial Services Benjamin Lawsky released sweeping new rules for licensing digital-currency businesses in the state Wednesday, staking part of his legacy on launching a specialized regulatory regime for an industry that many experts believe could play a significant role in the financial system.

The new rules, which Mr. Lawsky announced during a speech in Washington—his last as superintendent before departing at the end of this month—were met with a mix of condemnation and support from...