I think humanity's reaction will vary greatly on what type of life or data we discover.
For example, if we find evidence of past microbial life on Mars the scientific community (myself included) will be beside themselves with shock and enthusiasm. But I predict the general public will only be vaguely impressed. Many people probably already believe in "Martians" and don't know what a fossil or a microbe is.
Likewise if we discover microbial life anywhere outside of Earth, even if it's currently alive and not ancient fossils, I think most of humanity will be unimpressed and skeptical that it exists.
But I think things will be very different if we discover evidence of high-order intelligence and/or technology. For example, if Tabby's Star turns out to have a real alien Mega-Structure around it, the possibility of us learning new physics from merely observing how it is constructed will present us with the imperative of gaining any of that knowledge as fast as we can. And to that end, I think governments and businesses would quickly divert huge resources into research. Similarly, if we were to identify a signal of some type which contained information in any form, we would be inexorably drawn to it, not only for the implications to out knowledge of physics and technology, but also for simple entertainment. What would happen if we actually received "TV-like" images from an extraterrestrial source. I bet humanity would be glued to its screens for generations watching for any scraps of technology or information in the background. A situation like that would change humanity.