As much as I wish it were different, there is no way under the current interpretation of the First Amendment, to win a lawsuit against the newspaper. The press is given enormous leeway to behave stupidly.
In this case, the information was a public record and the newspaper published it. As reckless and stupid as it was, there wasn’t anything illegal about it. Yes, it was immoral, and inherently stupid. Yes, it was done with the malicious intent of maligning gun owners. However, the fact remains that newspapers are allowed to publish public records, no matter the negative consequences that might flow from their foolish decisions.
People think “sue” is a magic term that immediately brings showers of money along with unicorns and rainbows. That isn’t true. There has to be a valid ground for a lawsuit. Any lawsuit against the paper wouldn’t survive a motion to dismiss. The issue is with the New York law that requires handgun permits in the first place, and then makes that information a public record.
This may have actionable consequences if the plaintiffs appeal to the area of tort law called intentional tort. This is the legal theory that covers any deliberate interference with a legally recognized interest, such as the right to keep and bear arms. Other things such as the rights to bodily integrity, emotional tranquility, dominion over property, seclusion from public scrutiny, and freedom from confinement or deception would qualify. It is true that any jury would have to balance this against First Amendment issues, but applying a “reasonable person” standard, the “seclusion from public scrutiny” might carry some weight. The firearm owner is not a public figure, so the “absence of malice” defense would not apply. Remember that a unanimous vote of the jury is not required in tort cases.