Is there any way to stop the one-way brain drain from academia to industry?

You mentioned one answer (pay). There're others. I once communicated with someone who left his position as a professor at a major North American university. That surprised me since I knew the job was in high demand. I asked him why, and he responded:

Professor positions in major research university in US require generation of research funds - a process of writing your best ideas and sending them to funding agencies. The process of getting money from funding agencies is worse than the lottery. You spent a lot of creative effort in putting a proposal together and then most of the time it gets thrown into a garbage can. It is a general state of affairs, not just my experience. I just decided that I've had enough of that. When I spend time creating something I want it to see the light of day. So I switched fields and now work in a very dynamic industry, generating new knowledge or writing products that are actually used.

If any postdoc ask me for advice now - whether or not to go into academia - I would answer why would you torture yourself? There are so many fun jobs and are even better paying.

There's more. If a fresh PhD graduate stays in academia:

1. You live a nomadic lifestyle, hopping from one postdoc to another. This is not only bad for any significant others and children, but also a great hassle. Each time one moves, one needs new visas, needs to find accommodation, and so on. To add to that, postdocs offer no job security, and one is virtually always looking for a new job.
2. After that, if you're smart / lucky enough to get a permanent position, you have to generate research funds, which is unreliable (above).
3. After that, there's no guarantee you'll actually get tenure. If you are denied tenure, what are you going to do next? One is probably already >40 years old at that point.

Taken together, only the extremely passionate (or extremely masochistic) choose an academic career. For further reading I suggest these two articles which strongly shaped my view on this: Women in Science by Philip Greenspun, and Don't Become a Scientist! by Jonathan Katz.

Having said the above - why would a brain drain from academia to industry be a bad thing. It's simply market forces of supply and demand at play. If more people took the option to shift, there would be less competition for permanent positions. Less competition makes the academic path more attractive. Eventually things balance out. It's further possible for society to reverse the brain drain whenever it wishes, simply by providing more funding. If society doesn't want to do that, I don't see why academics should try to force it.