Immigration will likely be one issue Republicans will push next year as they seek to reclaim House seats in Republican-leaning New Jersey districts where Democrats won last year.

The state’s Democratic attorney general, Gurbir S. Grewal, first issued the Immigrant Trust Directive a year ago, but it did not take hold until March.

It laid out rules that blocked local and state law enforcement officers from asking people about their immigration status, or from helping federal immigration agents as they make detention arrests. It allows law enforcement officers to turn over undocumented immigrants charged with certain crimes to ICE agents, but only if those agents pick up the migrants on the day of their release.

The aim of the directive was to eliminate the fear of coming forward for people who are not in the country legally and may be victims of crimes, or witnesses who could be helpful to law enforcement authorities.

It came in response to the “overzealous enforcement environment that was driving people into the shadows,” Mr. Grewal said.

In Sussex County, the referendum was placed on the ballot by the county freeholders, at the request of the sheriff, who was re-elected on Tuesday. The state’s Republican chairman, Doug Steinhardt, helped to draft the wording that asked whether voters were in favor of county government providing support to ICE agents.

“It’s about putting public safety above politics,” said the Sussex County sheriff, Michael F. Strada. “The attorney general and the governor are endangering our communities.”