During testimony of a key witness in the fraud trial of Archegos Capital executives, Judge Alvin Hellerstein abruptly paused the proceedings to take a FaceTime call.

Holding up his phone, Hellerstein proudly showed the Manhattan federal courtroom his newest—and seventh—great-grandchild.

“The jurors exclaimed, ‘Oh my gosh, oh my gosh,’” recalls Jordan Estes, one of the lawyers on the case. Afterward the jury gave the judge a congratulatory card.

Now, at 92 years old, Hellerstein is presiding over the virtually unprecedented narcoterrorism case against ousted Venezuelan President Nicolás Maduro. An experienced jurist known for his unconventional approach to the job, the judge’s age is nonetheless raising questions about his ability to oversee a sprawling matter that could drag on for years.

Shira Scheindlin, a former district judge who served with Hellerstein, called him an unflinching jurist who has done what “he thought was right in every case, whether it was popular or not.”

Still, said Scheindlin, it is fair to raise questions about the age of any judge. “You don’t want any perception that the trial judge in any high-profile case wasn’t 100% at the top of his or her game. That would be a bad thing for the world to see,” she said.

Hellerstein was raised in the Bronx and earned his law degree at Columbia University, before spending nearly four decades in private practice at now-defunct Wall Street law firm Stroock & Stroock & Lavan. President Bill Clinton nominated him to the bench in 1998.

Hellerstein, an Orthodox Jew, has said that he schedules sentencing hearings for Fridays so he has all of the Sabbath to reflect on his decisions—and wrestles with them even afterward.

“I never, ever have had the feeling that my punishment was just and measured,” Hellerstein said on a podcast in 2020. “And that’s good. I think a judge should be disconcerted by what he does.”

He is known for presiding over decadelong tort litigation involving more than 10,000 cleanup workers and responders to the terrorist attacks of Sept. 11, 2001. In a move that was criticized at the time, Hellerstein rejected a proposed settlement—even though both sides agreed to it. He said the deal provided too much money to lawyers instead of their clients.

In 2020, Hellerstein released former Trump lawyer Michael Cohen from prison in response to a civil lawsuit, saying the Justice Department retaliated against him by revoking his home confinement. The jurist is also overseeing President Trump’s continuing bid to move his hush-money case from state to federal court.

Lawyers who have practiced before Hellerstein say he can be charming toward jurors but is tough on prosecutors and defense attorneys alike. An active participant in proceedings, he at times asks questions of witnesses himself, and is known for occasional informality.

During one court conference with just lawyers and a few others present, the judge never took his seat at the bench, recalled former prosecutor David Raskin. “He was wearing a tweed jacket and looked like a professor,” said Raskin, now at investigations firm Nardello & Co. “He was sitting at the counsel table, and we had the conference there.”

Hellerstein is also known for the unusual practice of calling lawyers individually on the phone to ask questions about cases. In the case of a New York real-estate developer, an appeals court admonished Hellerstein for calling a prosecutor. The appellate court said communications in criminal cases should generally be on the record.

While it isn’t common for judges to work into their 90s, Hellerstein is part of a small group of sitting federal judges who do. There are 64 out of more than 1,300 federal judges of that advanced age. Most, including Hellerstein, have taken senior status, meaning they usually carry a lighter caseload.

There is no mandatory age for federal judges to retire. One of Hellerstein’s fellow Manhattan jurists, Louis Stanton, is 98 years old, and a Brooklyn judge, Leo Glasser, is 101.

If Hellerstein were to become unable to see the case through because of either illness or death, it would be reassigned to another judge in the Southern District of New York.

“At the very beginning of the process another judge can step in fairly seamlessly, but that just becomes much more complicated the further we are in the proceedings,” said Marin Levy, a law professor at Duke University.

Hellerstein’s friend and former law partner, Joel Cohen, said the judge wouldn’t have decided to stay on the case without his usual soul-searching. “I’m quite confident he has deeply considered his capacity to undertake this important responsibility,” Cohen said.

Senior judges have to show the chief judge of their circuit that they are handling a certain amount of work annually, and concerns about their fitness could be raised during that process. Lawyers can also argue a judge is mentally unfit because of age, but legal scholars say it is hard to prove and typically only happens on appeal. It can sometimes take an aging judge’s own colleagues to convince the judge it is time to step aside.

Judge Pauline Newman, who sits on a federal appeals court that oversees patent cases, was suspended in 2023 after court staff raised concerns about the then 96-year old’s cognition. She has unsuccessfully been fighting her suspension ever since.

Lawyers who have appeared before Hellerstein say he can be curious and engaged, although at times becomes abrasive by the afternoon of a long trial day. For at least the past several years, lawyers say, he has appeared at times to nod off on the bench. (Lawyers and federal judges noted that Hellerstein is far from the only jurist to close his eyes—or even snooze—in court.)

Arthur Hellman, a University of Pittsburgh emeritus law professor who studies judicial ethics, said there are inherent risks in having any 90-something judge preside over a lengthy case.

“We’re not just talking about his condition today,” said Hellman, who is himself 83. “When you’re 92, people can decline very fast.”

Write to Corinne Ramey at corinne.ramey@wsj.com, Lydia Wheeler at lydia.wheeler@wsj.com and C. Ryan Barber at ryan.barber@wsj.com