One of the reasons I liked my physical/biological anthro class so much this semester is because it stressed the importance of having some degree of scientific literacy. I knew that was important to have, but I never really realized to what extent. And I myself wasn't-- and still am not really-- scientifically literate.
There's a lot of benefits to knowing at least a little bit about how science works. I think mainly, if nothing else, it's important to know how the scientific process works. As in, while its aim is to be objective, it doesn't always succeed in that. It's also correctable; what may be the accepted theory now could change drastically, as more tests are done and new technology aids scientists in figuring things out. Science also strictly applies to things that can be tested/repeatedly retested and (dis)proven. So if there's no real way to test it (i.e. palm reading, ghosts, creationism), it's not scientific.
(That doesn't mean belief in those things is wrong, by the way. That just means it's incompatible with science-- for instance, my teacher compared religion and science to oil and water.)
The important thing to take from that is how to distinguish science from pseudoscience. The ability to do that ensures that you'll most likely be getting the most accurate information available at the time, rather than fall prey to misinformation, which often relies on fear-mongering.
The other important thing is that you realize how important knowing even a little bit about science is. It can inform your opinion on a wide variety of topics-- anything from vaccinations, to GMOs, to global climate change, to sex ed. So many of the decisions we make have something to do with science, and knowing how to educate yourself and get the most accurate information in order to make the best decisions is immensely important.