Reagan announced the Strategic Defense Initiative in a major public address on March 23, 1983. It was widely covered by news stations and newspapers, it quickly was both attacked and defended in the popular media. It was one of the big issues of the early-to-mid 1980s. So — yes, pretty common knowledge.

The program was hugely costly, and Reagan had to sell the American people on it. It could not be conducted in secret. The details, of course, could be secret. But the fact that a major program was going on was not only something Reagan had to tell the people, but something he wanted to tell the people. Reagan really believed that this was the "way out" of the nuclear dilemma, and he honestly could not understand objections to it. (When he met with Gorbachev at Reykjavik in 1986, he even offered the technology to the USSR. Gorbachev pointed out that the US would not even give the USSR the technology the milk cows — he could not really believe they would give them missile defense.) It was a major selling point of his presidency in his mind. As he wrote in his diary, soon after hearing about the idea and deciding to push forward with it on a scale that none of his advisors anticipated:

"[It was] a super idea. So far the only policy worldwide on nuclear weapons is to have a deterrent. What if we tell the world we want to protect our people, not avenge them; that we're going to embark on a program of research to come up with a defensive weapon that could make nuclear weapons obsolete? I would call upon the scientific community to volunteer in bringing such a thing about."

More on Reagan and SDI can be found in David Hoffman, The Dead Hand.

I'm seeing some anecdotal evidence here that it was indeed the case. However it seems strange to me that such matters of defence would be talked about so openly during such a turbulent time, is there any evidence to suggest it was just a show for the Soviets? Or perhaps a way to garner public support?