World overpopulation is one of the biggest problems posed by the eventual development of biotechnologies that can reverse aging. People who now die from old age will stay around. Since their entire bodies will be rejuvenated they'll also regain or maintain the ability to reproduce. The death rate will drop to a level caused by accidents, murder, and suicides. At the same time, more people will have babies and they'll be able to have babies for hundreds and perhaps even thousands of years.
What to do about it? I have one thought: Genetically engineer offspring to remove instincts for reproduction. Surely instinctive desires for children exist in our DNA and they'll be discovered within the next couple of decades as DNA testing technologies become cheap and easy. So in theory it should be possible to make sure that at least new generations will not desire to have kids.
The people who are born to lack the instincts to reproduce won't miss these desires. Imagine you lacked the desire to eat some kind of food and found the taste of it disgusting. Would you miss it? I know even women who are totally turned off by the idea of having kids. They do not miss that desire at all.
The vast majority of people who are already alive when we develop rejuvenation technologies pose a much bigger problem. Faced with centuries of life and driven by instincts that make them like babies and children they'll want to have kids. Some will want to have more kids after their own have long grown up and moved away. Many want grandchildren and so will oppose genetic engineering of offspring to remove the reproduction instincts. So it is not clear that my proposal here has a chance. But I figure it is worth some discussion.
One final point: Genetically engineering people to lack the reproduction instinct? Are you mad? Well, I'm already imagining a world where most people (at least in industrialized countries) do not grow old. People will use drugs to enhance their intelligence and control their emotions. They'll genetically engineer their offspring to look and think more to the liking of the parents. These are all things most will embrace voluntarily as soon as it becomes possible to do so. But we are driven by instincts as much as by reason and some of our instincts have become quite problematic for the health of society. In the interest of freedom should we allow people to act on their instincts and drive the human population up into the tens of billions? Sounds like a bad future to me. We should find some way to avoid it.
Update: To the people who argue that fertility rates have already fallen in many countries and who expect that trend to continue I say you underestimate the power of natural selection. The fall in fertility rates is exerting a strong selective pressure for genetic variations that increase fertility. Those people who are having babies are passing along more alleles that favor reproduction in industrial societies than existed in previous generations. New generations of the human race are getting selected to have greater desire to have kids. Whatever genetic qualities which increase desire to have kids are getting selected for. What desires and attributes which distract from having kids are getting selected against.
The human race has to defeat Darwinian natural selection if it is to prevent runaway population growth. The only way to defeat selective pressures for reproduction in an eternally youthful society is to genetically reengineer humanity to reduce reproductive urges. Selective pressures for reproduction could be delayed and reduced in a society ruled by a totalitarian government that forbade reproduction. But such a government would not be stable because a very sizable fraction of the population would feel frustration and anger over restrictions on reproduction and would seek to overthrow the government.