In 1938, the Great Depression and the looming danger from Europe (combined with memories of World War I) created a social situation in which Orson Welles' broadcast could seem more believable and frightening.
At this point in history, people felt very vulnerable. They had been scarred by almost a decade of economic downturn. They would have felt that the evidence of World War I showed that technology could be used not just for good, but also to create horror and devastation. They would have been on edge as Hitler pushed Europe to the brink (at that point) of another war in which technology would be used in negative ways. In this context, it makes sense that Welles' broadcast would have had a more powerful impact than it might have at another time in our history.
Welles' broadcast would have played into the fears that people were feeling. It would have confirmed in them the idea that they were vulnerable to forces (economics, Hitler, and now Martians) that were completely beyond their control. It would have seemed plausible because it would have been clear to them that technology could be used to devastate in the way the fictional Martians were using it.
In these ways, the fears and experiences of people in 1938 would have helped to make Welles' broadcast seem more credible and to make it have more of an impact on people than it might have in better times.