“I fundamentally don’t believe that the Northeast will solve our young people shortage until we become a more welcoming place for people who are newcomers to America,” said Gov. Peter Shumlin, Democrat of Vermont, a state that is almost as old and white as Maine and is caught up in its own controversies over refugee resettlement.

Many immigrants have found welcoming communities and do not plan to leave, despite Mr. LePage’s antipathy toward some foreign-born newcomers. Still, the state’s demographic challenges have set off alarm bells.

A new report from the Maine Department of Labor says the work force is shrinking just as demand for certain jobs, particularly in health care, is increasing and leading to shortages that will ripple across the state.

“That translates ultimately into a health care crisis, a work force crisis, it will affect the whole notion of what small-town life is like,” said State Representative Erik C. Jorgensen, a Democrat who represents Portland, one of the few cities in Maine that has generally welcomed immigrants.

Like many states in the Northeast, Maine has had difficulty retaining young people and attracting those from other states. Its relatively low wages, high cost of energy, remoteness and long, harsh winters may account for some of the resistance.

“A lot of the challenge is making Maine a place where people want to be and are able to make a living and trying to compete with places like New Hampshire, which is very similar to Maine but doesn’t have an income tax,” said Ms. Rector, the state economist. (Mr. LePage has sought, unsuccessfully, to eliminate Maine’s income tax.)

Whatever the cause, many businesses and local governments are struggling to find qualified workers. Maine is projecting a drop to 2.1 people of working age per retiree by 2032, down from 3.2 per retiree in 2017, according to the Maine Development Foundation, a private organization devoted to improving the state’s economy.