First of all, let us identify some of the main reasons why public trust in science gets undermined.
Science is complicated. It often requires specialized knowledge and experience to grasp the significance of scientific advances and to appreciate their practical implications. People fear what they do not understand. We all believe in better public education about science, but our ideas about how to promote it may be unrealistic. It is implausible that there will be big changes in public "scientific literacy." We need to learn to do better talking to people where they are. The challenge of doing so is great. Many scientists live in a subculture that is quite detached from ordinary life, even from the ordinary life of educated members of affluent societies. With other groups, the cultural gap is even larger.
Scientists overpromise. There is a widespread belief, not entirely without foundation, that the most effective way to "sell" science is emphasizing near-term practical applications, particularly to human health. History teaches us that advancing scientific knowledge does correlate, on a time scale of decades, with applications that greatly improve human welfare. On a shorter time scale, the relationship between support for science and practical applications is highly unpredictable. Even the relationship between major scientific discoveries and practical applications is erratic. We live in an impatient, utilitarian age in which cautious voices are readily drowned out by hucksterism. Indeed, it is age in which hucksterism pays particularly well.
Scientists are losing their scholarly aura. Along with high salaries, large grants, lucrative consulting fees, and stock options comes loss of credibility. Many scientists, particularly those with high public profiles, are serving a variety of masters. When they speak out on issues of public interest, it is not always clear that they are bringing their best, objective judgment to bear on complex issues. They may simply be trying to run up the stock price of their company.
Scientists do not always bring good news. Humanity is facing problems of ever-increasing complexity. Scientists are often the source of the unwelcome news that these problems are real, that some of them will get worse before they get better, and that politically popular "solutions" to them are poorly considered. Shooting the messenger who brings bad news will remain in style.