The Reagan connection to WarGames likely stems from an anecdote from Lee Hamilton, a Democratic Congressman from Indiana and Vic Fazio from California. Hamilton and Fazio had been two of the Democratic voices backing the MX program and had went to the White House with other congressmen for Reagan to give his assurance that he would make a serious commitment to arms control if congress supported the MX missile.

According to Hamilton, Reagan gave standard boilerplate answers to the nuclear issue, often reading from cue cards. In middle of a complex presentation on the MX, the President then asked the congressional delegation if any of them had seen WarGames, at which point the President became very animated as he had seen the movie two days before. Fazio recalled in a later interview:

Reagan suddenly asks us if we have seen WarGames. He was in a very good humor. He said “I don’t understand these computers very well, but this young man obviously did. He had tied into NORAD!” Reagan then continued with his impromptu review by saying he found a little bias in casting the high school teacher in the movie as “a wimp.” The he turned to Army General John W. Vesey Jr., chairman of the Joint Chiefs of Staff, and then said with a smile, “They portrayed the general as this slovenly, mean unthinking guy.”Vassey’s face reddened.

These accounts have to be taken with some degree of salt as they are somewhat partisan and this one instance of WarGames doesn’t preclude that Reagan had multiple takes on WarGames. Reagan was quite fond of using film and television to understand or explain concepts. By many accounts, he was deeply shaken by the 1983 miniseries The Day After and that pushed him to support SDI. Lou Cannon, a journalist and one of Reagan’s biographers, observed of Reagan’s personality that:

All the lawyers who knew Reagan well…realized that Reagan took a different intellectual approach than they did. All of these lawyers also recognized that they could reach Reagan if they were willing to tune into his mental wavelength. The problem with Reagan’s reliance upon metaphor and analogy for understanding made him vulnerable to arguments that were short on facts and long on theatrical gimmicks.

Sources

Cannon, Lou. President Reagan The Role of a Lifetime. New York: PublicAffairs, 2008.

Fischer, Beth A. The Reagan Reversal: Foreign Policy and the End of the Cold War. Missouri, Mo: University of Missouri Press, 2000.

Two thoughts:

• The story I have usually heard is that Reagan was very much affected to opposed nuclear weapons after watching The Day After, not WarGames. This is something that can be also seen in his diaries from the time.

• There were several major glitches with NORAD's computers that happened in late 1979 and early 1980. Presumably Reagan would likely have been briefed on them at some point but I don't know that for sure; they did, however, end up in the newspapers not too long after they happened. From Schlosser's Command and Control:

AT ABOUT ELEVEN O’CLOCK in the morning on November 9, 1979, the computers at the NORAD headquarters inside Cheyenne Mountain said that the United States was under attack. The huge screen in the underground command center at SAC headquarters showed that Soviet missiles had been launched from submarines off the West Coast. The same message was received by computers in the National Military Command Center at the Pentagon and the Alternate National Military Command Center at Site R inside Raven Rock Mountain. And then more missiles appeared on the screen, launched not only from submarines but also from sites within the Soviet Union. It was a massive attack, and warheads would begin to hit American targets within five or six minutes.

Whenever NORAD’s early-warning sensors detected signs of a possible missile launch, a Missile Display Conference was held. It happened about four times a day; the infrared sensors on the Air Force satellites could be triggered by forest fires, volcanic eruptions, and other sources of heat. The officers on duty would discuss whether the threat seemed real or merely a false alarm. The commander in chief of NORAD would decide if a Threat Assessment Conference had to be arranged, bringing the head of SAC and the chairman of the Joint Chiefs of Staff into the discussion. That type of conference happened once or twice a week. And if missiles truly seemed to be heading toward the United States, a Missile Attack Conference would be set up. It would give the president a chance to speak with senior officers, listen to their advice, and decide whether to launch missiles in retaliation. A Missile Attack Conference had never been held.

As the computer screens at NORAD filled with Soviet missiles, a Threat Assessment Conference was called. Although the pattern of the attack seemed to fit with the Pentagon’s assumptions about Soviet war plans, its timing made little sense. Tensions between the superpowers weren’t particularly high, and nothing in the news seemed to warrant a “bolt from the blue” attack on the United States. Duty officers at NORAD contacted the radar and ground stations whose sensors were relaying information about the launches. None of them had detected signs of any missiles. The NORAD computers seemed to be providing an erroneous—but highly realistic—account of a Soviet surprise attack.

As a precaution, the Klaxons were sounded at SAC bases nationwide. Bomber crews ran to their planes, and missile crews were put on heightened alert. Fighter-interceptors took off to look for signs of a Soviet attack. The National Emergency Airborne Command Post left Andrews Air Force Base—without President Carter on board. And air traffic controllers throughout the country prepared to clear America’s airspace for military flights, warning every commercial airliner that it might soon have to land.

As the minutes passed without the arrival of Soviet warheads, it became clear that the United States wasn’t under attack. The cause of the false alarm was soon discovered. A technician had put the wrong tape into one of NORAD’s computers. The tape was part of a training exercise—a war game that simulated a Soviet attack on the United States. The computer had transmitted realistic details of the war game to SAC headquarters, the Pentagon, and Site R. [...]

“FALSE ALARM ON ATTACK SENDS FIGHTERS INTO SKY” was one of the headlines, when news of the training tape incident leaked. Pentagon officials denied that the missile warning had been taken seriously. But the technical and human errors at NORAD felt in keeping with the general mood of the country. An accidental nuclear war didn’t sound inconceivable to most people—America seemed to be falling apart. A few months earlier a nuclear reactor at Three Mile Island in Pennsylvania had suffered a partial meltdown, largely because a worker at the plant had turned off an emergency cooling system by mistake.

At about two thirty in the morning on June 3, 1980, Zbigniew Brzezinski, the president’s national security adviser, was awakened by a phone call from a staff member, General William E. Odom. Soviet submarines have launched 220 missiles at the United States, Odom said. This time a surprise attack wasn’t implausible. The Soviet Union had recently invaded Afghanistan, confirming every brutal stereotype promoted by the Committee on the Present Danger. The United States was leading a boycott of the upcoming Moscow Olympics, and relations between the two superpowers were at their lowest point since the Cuban Missile Crisis. Brzezinski told Odom to call him back with confirmation of the Soviet attack and its intended targets. The United States would have to retaliate immediately; once the details of the attack were clear, Brzezinski would notify the president. Odom called back and said that 2,200 missiles were heading toward the United States—almost every long-range missile in the Soviet arsenal. As Brzezinski prepared to phone the White House, Odom called again. The computers at NORAD said that Soviet missiles had been launched, but the early-warning radars and satellites hadn’t detected any. It was a false alarm. Brzezinski had allowed his wife to sleep through the whole episode, preferring that she not be awake when the warheads struck Washington.

SAC bomber crews had run to their planes and started the engines. Missile crews had been told to open their safes. The airborne command post of the Pacific Command had taken off. And then the duty officer at the Pentagon’s National Military Command Center ended the Threat Assessment Conference, confident that no Soviet missiles had been launched. Once again, NORAD’s computers and its early-warning sensors were saying different things. The problem was clearly in one of the computers, but it would be hard to find. A few days later NORAD computers warned SAC headquarters and the Pentagon for a third time that the United States was being attacked. Klaxons sounded, bomber crews ran to their planes—and another Threat Assessment Conference declared another false alarm.

This time technicians found the problem: a defective computer chip in a communications device. NORAD had dedicated lines that connected the computers inside Cheyenne Mountain to their counterparts at SAC headquarters, the Pentagon, and Site R. Day and night, NORAD sent test messages to ensure that those lines were working. The test message was a warning of a missile attack—with zeros always inserted in the space showing the number of missiles that had been launched. The faulty computer chip had randomly put the number 2 in that space, suggesting that 2 missiles, 220 missiles, or 2,200 missiles had been launched. The defective chip was replaced, at a cost of forty-six cents. And a new test message was written for NORAD’s dedicated lines. It did not mention any missiles.

Sorry for the long cut-and-paste, but I thought it was worth reading in the original. Schlosser's book is great if you are interested in stuff like this.