Tyler Cowen's essay was mentioned here before and I think it's a pretty good way to frame your thinking about the subject. The reason the prestige of science and engineering declined is because the direct and visible impact on every day life of those professions has also declined. A lot of technical areas have matured since the 1950's and are offering mostly incremental improvements.
Not to say that the 'actual' impact has declined, but people in the United States are no longer impressed by a greener plastic or a slightly newer satellite being lofted into space. Compare that to places like China or even India that are somewhat behind in bringing industrialization to the entire population and you have engineers make visible impact to every day life and therefore rationally have more importance and thereby prestige. I'm sure that in a few decades once they start catching up, engineering will also start declining as a prestigious profession.
There are lots of other intellectually demanding professions such as law, medicine and even finance that have droves of bright students signing up. I think that while they attract students, and are useful at all stages of development, they really come up to the top once you have a stable political system, and a relatively wealthy population. Once technical change stops being radical you enter a maintenance phase. If you have basic health needs met you start trying to squeeze out more and more advances in medicine to prolong the life of a relatively healthy population. Once new products become mostly enhancements of old ideas, trying to compete using the law can be more efficient in terms of effort expanded to results achieved.
I think that trying to use marketing or propaganda to try and raise the prestige of scientists can only go so far. Technology needs to offer radical and easily visible everyday impact to outshine the other intellectual professions.