I think there are several reasons (mostly misconceptions) and we need to tackle those. Here's a first attempt:
The young may not be entranced by glitzy lifestyles on TV (at least, I don't think the majority are) but earning a decent living is still a concern. Well, science jobs can't compete with the private sector but they do pay pretty well (and they're linked to other government pay scales and come with the other usual benefits). All the scientists I know live comfortably and enjoy life. And few other careers give you the same job satisfaction and independence.
- Availability of jobs
If you're good, you'll get a job. All the top research institutes are expanding and eager to attract new talent. If you think, halfway through your Ph.D., that a research career is not for you, you can exit any time and get a good job -- the training will be useful. Physicists in particular have excellent job opportunities in financial institutions and other sectors that require a talent at mathematical modelling. (Note to anyone from IISc who's reading this: IT'S OK FOR STUDENTS TO LEAVE. Even if they're leaving to do a PhD elsewhere. Don't shut down programmes like the integrated Ph.D. on those grounds. First get the good students in, and then motivate them to stay.)
- Work environment
Many people, ignorant of science in India, asked me when I moved back here: "Isn't the work atmosphere stifling? Don't you have to do what the boss tells you? Aren't there layers of bureaucracy?" No, no, and no. There is no boss, facilities are excellent, computer systems are the best, best-maintained, and least-bureaucratic I've seen anywhere in the world. My workplace may be particularly good, but this seems to be true at all the places I've seen.
Though faculty members don't have bosses, students (and, often, postdocs) do have de-facto bosses -- their faculty supervisors. And it is true that student-advisor relations span a broad spectrum, from excellent to absolutely awful. Still, I know students who hated their advisors' guts but enjoyed the science and went on to have good careers. A doctoral student's life is probably much more enjoyable than an apprenticeship in any other field.
- Scientists are boring
No. Software engineers and business executives are boring. Scientists aren't.
- The mating factor
Stemming from the above, this is something of an international worry (see this post for example). Don't worry, science students come in both genders, and even if you don't meet your match in the lab, there are enough bright non-scientists who are attracted to scientists (of both genders).
Anything I left out?