The Brooklyn borough president, Eric L. Adams, a leading critic of police methods in the wake of the Garner case, said Mr. Gentile was wise to avoid any appearance of exploiting the issue. Mr. Adams, a Democrat who plans to campaign for Mr. Gentile, said he had advised the councilman to focus on grand jury secrecy.

“The case has probably helped Donovan more than hurt him,” said Mr. Adams, noting that the issue tended to be particularly divisive. “Those who dislike the decision not to indict are passionate about it, but on the other side of the coin, people are passionate that the officer should not have been indicted.”

Still, some Democrats consider it a mistake to tread so lightly around the grand jury matter: Mr. Sharpton, who emphasized that he has not been engaged in the race, said Democrats risked depressing turnout by ducking an issue of deep concern to many in the party’s base.

Mr. Sharpton compared Mr. Donovan unfavorably to the authorities in North Charleston, S.C., where a police officer was swiftly indicted and dismissed from his job this month after shooting a fleeing, unarmed man in the course of a routine traffic stop.

“The contrast between him and a police chief in a state that still flies a Confederate flag on the Capitol is something that independent voters may be concerned about,” Mr. Sharpton said. “In a special election, you can’t go by the normal political demographics. You have to go by turnout — who energizes their base to come out.”

Mr. Gentile’s hope is that, even with the Garner issue having faded, Democrats will show up to vote based on his feisty economic message. He has defended the Affordable Care Act (Republicans, Mr. Gentile said, aim to “take 16.4 million people off Obamacare, and they offer no alternative”) and announced his opposition to the Trans-Pacific Partnership, a trade agreement the White House backs and many unions oppose.

Image Lisa Kogen, a Democrat, called the Garner case “appalling” but understood Mr. Gentile’s tiptoeing around the issue. Credit... Michael Appleton for The New York Times

But Mr. Gentile acknowledged that turning out voters would be a challenge, and said it would be easier to win the seat in a presidential election year, with a national candidate like Hillary Rodham Clinton driving participation. He faces another disadvantage: Mr. Donovan has raised more money, outpacing Mr. Gentile three to one, Federal Election Commission records show.