First and foremost is the quality of science education, starting in elementary school.
The second is to be sure that science itself does not seem to have a political colouring that would turn off the part of the population from a different part of the political spectrum. If science is seen as a left-wing activity, people on the right will blow it off. That's probably the main reason. In fact, that is certainly the main reason for rejection of scientific consensus in the studies that I have seen. It is not because of scientific ignorance; it is because of perceived ideological contamination of science.
The third is not easy for governments to manipulate, and that is cultural norms as to whether science is an attractive career option. That depends on peer influence, on culture and on many things that government policy may not easily be able to control directly: Is science cool?