Interesting question. I'm reminded of the social convulsions and existential angst which A.C. Clarke and Gentry Lee described at the start of
Rama II.
Personally, I think they got it all wrong. I don't think the abstract knowledge that somewhere out there, basically beyond our reach, there are other intelligent beings, would bring much of a change for most people. It would be exciting for space enthusiasts like us, but most people would just stop for a while to take note, and then quickly move on with their lives as the novelty wore off. Just think of the reaction to the discovery of the hobbit-man, last year.
I suppose there would also be those who, as usual, would either:
a) Claim the discovery was a hoax manufactured by the scientific establishment.
b) Claim that there was much more information about the aliens, but it was being withheld by the scientific establishment.
Speaking of Clarke, another thing I always felt he got wrong, namely in
The Fountains of Paradise, is the idea that discovering intelligence elsewhere in the universe would be a blow to religion. I don't think it would be more of a blow to religion than the discovery of Uranus was to astrology.