Democrat Phil Murphy cruised to victory in New Jersey’s gubernatorial race in November. The state’s powerful public-sector unions, which endorsed his progressive vision of higher taxes and more spending, played a pivotal role in his 14-point victory. Now the hard part begins.

As he takes office later this month, Mr. Murphy must confront the state’s biggest problem—a pension system that is about $90 billion short of what it needs to pay future benefits. He has already promised to devote some of the new revenue from his proposed...