But there isn't a singular date to associate with the Red Scare... that's more a nebulous period of time. It was also a gradual process. The West was already preparing for the oncoming cold-war with the USSR before the end of WW2.
The loyalty boards and Hollywood blacklisting were definitely a change is society, but it was so slow moving that I wouldn't compare it to a shift like 9/11.
The world wasn't really different before and after something like the hearings of the House Un-American Activities Committee, even though that is what would be a likely event you'd associate with the Red Scare.
When I think of 20th century cultural events that had an shift in society, I think more of the Cuban missile crisis, the Challenger disaster, the fall of the Berlin wall, etc... shorter events (not necessarily instantaneous, but well defined times) where you could define shifts in culture to before/after.
During the Red Scare, loyalty review boards were instituted and civil servants insufficiently 'American' were sacked in the thousands.