It's nearly impossible to imagine just how big of an event it would be if SETI found a message from another species. Intelligence, somewhere else in the universe, calling out to us in the darkness. It would change everything. Similar to the first images of the earth from space and the first understanding that we were not the center of the universe this is an event that would change our entire perspective of existence in ways that are difficult to imagine.
The message would almost certainly take quite some time to decipher. It would not only be in a language we didn't understand but likely using a type of mind that we don't understand. Yet even before we knew what the message the likely first reaction of many people would be fear. Even though the message would likely be from someplace many light years away with no connection humans first reaction to change is often fear.
That fear wouldn't last long unless there was something odd in the message though. After a few months people would mostly stop being afraid. There would still be some religious people. Who were convinced that the message was from demons but most religious people wouldn't be all that effected.
What would be likely be effected would be new and old religions who wrote the message into their doctrine. Even before the message was translated new religions would likely form. Humans desire to connect to others and the desire to connect to the supernatural working together, as they become the center of new cults.
It is impossible to really know what would be in a message, but assuming it was a simple message with no real scientific or cultural insight on the creatures the message itself wouldn't matter all that much. Instead it might convince a few people that we have more in common than we thought. Along with so many other effects of the modern world it would serve to make the earth a little smaller.
Massive spending would likely begin in improving the search. We would examine that world carefully, the best telescopes we can build would be aimed at it. Messages may be sent that reply to the first. Money would begin to pour into NASA and the space program would begin to move forward faster than it has since the sixties. With the full understanding that there really is something out there our natural instinct to explore would become stronger.
If the message itself didn't contain some profound wisdom or understanding of the universe it is in the end unlikely that any of this change would remain as earth shattering as it first did. Like landing on the moon it still a vital point in our history, but the culture largely stayed the same. Humans are a species that doesn't change quickly but this knowledge would slowly change us.