His move against unionized state workers is drawing even more attention as states across the country consider how to respond to a Supreme Court decision last year that limited the power of unions to collect fees from nonmembers.

The escalating battle in Alaska recalls a much bigger one by former Gov. Scott Walker of Wisconsin, who rose to prominence in 2011 as he championed a set of laws that undermined labor organizations and set off a precipitous decline in the state’s union rolls before losing re-election last year.

Union membership has been declining for decades, in part as a result of wider moves to make it harder to sign and hold dues-paying members. Legislation to impose additional restrictions on union enrollments was proposed this year in Montana, Kansas and Pennsylvania.

Alaska’s plan has gone the furthest. Under Mr. Dunleavy’s new policy, state workers will have to declare that they want to opt into the union and sign an acknowledgement that they know they do not have to have such representation. They will then need a second layer of authentication, such as an email exchange, to reaffirm their intentions.

And they will have to repeat the opt-in process every year.

Mr. Dunleavy said the plan was intended not to harm unions, but to comply with the Supreme Court, which ruled last year that government workers who elect not to join the union do not have to help pay for collective bargaining.