Not only is the scientific mindset absolutely essential for performing good science, but it's also advantageous to cultivate this way of approaching the world in day-to-day life.
Curiosity is the heart and soul of science. Scientists never lose that childhood infatuation with the question "Why?". This single question has guided many of our discoveries and has led to much of our understanding of the world around us.
A scientist approaches the universe with an insatiable curiosity. We look for patterns in the world around us. We ask questions of "why" and "how" and "what if" about everything we see. It's not enough to know something is; we want to know why it is, how it became that way, and whether it will ever change.
An Open Mind
A good scientist always keeps an open mind. When beginning to attempt to answer the many questions her curiosity brings to light, she considers ALL possibilities. She relies upon the scientific method to separate the true from the ridiculous. In the beginning, all possibilities are valid.
Despite what many among the ignorant think, science is NOT about having a "pet theory" and designing experiments to "prove" that theory. That's called "pseudo-science". A good example of pseudo-science is the load of horseshit called "Intelligent Design". (Don't take my word for it, the teaching of intelligent design was ruled unconstitutional in the Kitzmiller v. Dover case. Learn more in this excellent NOVA program.)
Rather, a scientist seeks to DISprove a hypothesis through experiment. Experiments can only disprove a theory, nothing is truly ever proven in science, because there's always the possibility of new data and understanding modifying a hypothesis. This happens all the time in science as our understanding grows. For instance, it was once believed that nothing could escape a black hole, but now it's believed that black holes actually emit what's called "Hawking radiation".
Many scientific theories that are now accepted and widely used in technology were originally discounted by many as "crazy". For instance, Einstein's idea in special relativity that time doesn't flow at the same rate at all places in the universe--that the flow of time is affected by gravity and velocity. We now know this to be true, even though people once thought it was crazy (because it defied "common sense", or as I like to call it, "common nonsense"). In fact, GPS works based on this principle. It's because of the variable rate of time that the GPS in your car or phone can tell you exactly where you are.
A scientist's open-mind must be tempered by a very healthy dose of skepticism. While in the beginning of the experiment all possibilities must be considered, eventually we must start weeding out the erroneous ideas in our quest for truth. A maxim in science is "correlation is not causation". Scientists must always scrutinize their results and conclusions. They always search for hidden variables--things that may be producing the observed effect that are outside what they've considered.
Most of us know people who believe everything they've heard or read, no matter the source. A good scientist always checks for the sources of information. Every piece of data must be scrutinized. Unfortunately, as humans we love "proving ourselves right". Skepticism combats this tendency, which can often blind us to evidence that we're not right.
All scientific findings are open to peer-review. This is another hallmark of science that many among the ignorant don't understand. Every published paper is open to debate and debunking. A proper scientific paper gives the reader all the information they need to carry out the experiments themselves, and many do. Many of the lovers of conspiracy claim that science is "bought". While it's true that any given scientist's ethics may be compromised by large sums of money (we're only human, after all), the peer review process will weed these out pretty quickly. Scientists are harsh critics, and someone exposed in such a way can kiss his career in science goodbye.
Finally, a good scientist is humble. In the search for truth, one must be able to admit when one is wrong. One must be willing to throw out a hypothesis that doesn't hold up to testing. It may seem counter-intuitive, but most scientists are actually happy when an experiment proves a hypothesis wrong. The reason is that this is the way the secrets of the universe are teased out. A hypothesis can never be proven with 100% certainty. Theories in science are those hypotheses that have been supported by countless experiments over years and years of study, but even these theories could be disproven with more sophisticated knowledge in the future (or at least amended, which happens all the time). However, a hypothesis CAN be proven with certainty.
Before science, people answered their questions with superstition. People lived in fear of the unknown, the unknowable. To combat that fear of what hid in the dark, man invented myth, superstition and religion. Today, we have a better way of figuring out the universe. It's true that some people still choose to live in ignorance. But they have the CHOICE, and choice is a beautiful thing.