Lowering birth rates by increasing peoples' standard of living is one of the few climate change wins we can plausibly attribute to human nature. Don't need any of that eugenics shit, just provide people the opportunity to lead good dignified enriching lives.
I don't trust these attempts under capitalism one bit, but a well-functioning state probably has an interest in not seeing its lonely people driven crazier than necessary by algorithms designed to waste users' time and make money. I used one like 10 years ago or more but all I hear is how horrible they are from everybody I know. Either ban it or make it functional, I say.
As climate change makes more of the planet uninhabitable, people will be forced to move to the habitable zones, increasing population density in those places. Assuming we don't get to total extinction of humans, we'll eventually hit the floor for birth rates and they'll start creeping up again. Might be a few centuries before things stabilize enough for that to happen, though.
That's assuming the economic arrangement during and post climate change is conductive to rising birth rates in the first place.
What people, especially on the right, don't realize is that falling birth rates is not a matter of incentive or culture. It's correlated with rising literacy rates. That's it. The moment people realize the costs of raising a child in an industrial society and how that affects their future livelihoods they simply don't. The reason birth rates were high in pre industrial societies is because having lots of children was the economic strategy of every living human - from peasants to landlords to kings and merchants. The reason we had exploding growth rates during the industrializing period is because literacy rates and standards of living actually went down in many places, and didn't catch up to the industrial reality until much later. Now, having a child is a luxury for the gentry few. Therefore, populations de-grow.