Dukakis' coolly technocratic response was widely seen as damaging to his campaign, and Shaw said later he got a flood of hate mail for asking it.
"Since when did a question hurt a politician?" Shaw said in an interview aired by CSPAN in 2001. "It wasn't the question. It was the answer."
Shaw memorably reported, with correspondents Peter Arnett and John Holliman, from a hotel room in Baghdad as CNN aired stunning footage of airstrikes and anti-aircraft fire at the beginning of the US invasion to liberate Kuwait.
"I've never been there," he said that night, "but this feels like we're in the centre of hell."
The reports were crucial in establishing CNN when it was the only cable news network and broadcasters ABC, CBS and NBC dominated television news.
"He put CNN on the map," said Frank Sesno, a former CNN Washington bureau chief and now a professor at George Washington University.
Shaw, who grew up in Chicago wanting to be a journalist and admiring legendary CBS newsmen Edward R. Murrow and Walter Cronkite, recognised it as a key moment.
"In all of the years of preparing to be anchor, one of the things I strove for was to be able to control my emotions in the midst of hell breaking out," Shaw said in a 2014 interview with NPR.
"And I personally feel that I passed my stringent test for that in Baghdad."
Shaw covered the demonstrations in China's Tiananmen Square in 1989, signing off as authorities told CNN to stop its telecast. While at ABC, he was one of the first reporters on the scene of the 1978 Jonestown massacre.
On Twitter, CNN's John King paid tribute to Shaw's "soft-spoken yet booming voice" and said he was a mentor and role model to many.
"Bernard Shaw exemplified excellence in his life," Johnson said. "He will be remembered as a fierce advocate of responsible journalism."
CNN's current chief executive, Chris Licht, paid tribute to Shaw as a CNN original who made appearances on the network as recently as last year to provide commentary.
So guarded against any appearance of bias that he didn't vote, Shaw asked tough questions of several politicians. He asked George H.W. Bush's pick for vice president, Dan Quayle, if "fear of being killed in Vietnam" led to Quayle joining the National Guard in 1969.
As a member of the US Marines, Shaw angled for a meeting with one of his heroes, Cronkite, in Hawaii in 1961.
"He was the most persistent guy I've ever met in my life," the late Cronkite told the Washington Post in 1991.
"I was going to give him five begrudging minutes and ended up talking to him for a half hour. He was just determined to be a journalist."
He got a radio job in Chicago, where an early assignment was covering an appearance by Martin Luther King. Shaw recalled for CNN King telling him, "one day you're going to make it. Just do some good".
In retiring at a relatively young age, Shaw acknowledged the toll on his personal life that went with being a successful journalist. Because of all the things he missed with his family while working, he told NPR that "I don't think it was worth it".
His funeral will be private, with a public memorial planned for later, Johnson said.
Shaw is survived by his wife, Linda, and two children.