It would be a matter of public education until at least a majority agreed that action is required.
Ok, but I think a lot hangs on the seemingly minor word 'until'.
Your idea here is predicated on the need for urgent, immediate action.
I can't envision a global wide education program that would encourage people to volunteer to have their children randomly sterilized. I expect that most people would - quite reasonably - imagine that there are superior ways to achieve the same desired end which don't involve infringing so dramatically on peoples' liberty.
However, let's imagine that over time people come to accept that it's necessary, but this means we're failing to address the key motivation supposedly justifying the infringement on peoples' freedom - the urgency of the need. How long would it take until a majority agrees? Generations? Isn't that contradictory to the urgency?
I agree that depriving only the selected of reproduction seems harsh, but how else can world population be fairly reduced to a sustainable level?
It's not that it's 'harsh' that's really the problem: it's that it's intrinsically unjust.
The word 'sacrifice' is quite insightful here.
On the one hand, if you believed strongly that you were helping others and elected to give up something important to yourself that it might benefit others, then your sacrifice is of the noble kind.
On the other, if you believe strongly that the need is sufficiently urgent to insist other people give up something important to them in order to benefit others, then the sacrifice therein is of the 'throwing human offerings into the volcano to appease the Lava God' kind.
Of course, as I've already pointed out - the idea that there is no other plausible means of addressing humanity's impact on the climate other than forcibly sterilizing random newborns is completely unwarranted - you have not remotely established that there are no alternatives, and I've just explained that there are already forces at play which could be harnessed, supported, and built on which don't involve battering a sub-section of humanity into compliance, but actually elevating them out of crushing poverty.
We need there to be less humans, but the way we get to that outcome is just as important as achieving it.
The traditional method is violence in the form of warfare or genocide or something chaotic like that.
Well, that's Malthusian. There's also pandemic, of course. The 4 Horsemen have long helped keep our numbers in check, but we're overcoming all of them to one degree or other. Our successes have contributed to the problem we're facing, but the bright side is that now we are aware of the impact we're having. Even a hundred years ago, we wouldn't have been having this conversation - people still believed that the Earth was basically boundless in resources and exploiting them was a noble pursuit, and that more children were a sign of divine favour. There are populations all over the world today who have reversed these ideas and their populations have plateaued or are in decline.
Letting a random number generator will at least give everyone a fair chance.
I don't conceive of your idea as having even so much as a modicum of 'fairness' to it. Quite the contrary.
Firstly, it entails forcing other people to take part in a system where they only stand to individually lose (what do they individually gain from partaking?), forcibly mutilating newborn infants who are unable to advocate for their own beliefs but who are having their lives decided for them before they even yet know what life is, deciding that a single idea which involves other peoples' sacrifice is the only idea without considering other less rash ideas... none of these are 'fair'. If I forced you at gunpoint to take part in a race where everyone but the winner would be murdered, I couldn't call it 'fair' just because the race conditions were the same for everyone.
Secondly, it employs a 'random number generator' presumably under the faulty assumption that random = fair. Random is not 'fair'. Fairness implies a sense of consideration, of weighing up the stakes involved and attempting to find a just outcome. Random is random, and is totally uncaring of any of the factors involved - it's abstracted outside of the problem. Random may well be impartial, but would that really be fair? For example, imagine that, by sheer chance, the last remaining members of an ancient tribe all happened to be randomly drawn, and thereby have their cultural and genetic lineage snuffed out. That could happen randomly, but I can't imagine how anyone would consider such an outcome 'fair'.
I find no means by which you could portray this idea as fair. I think it is intrinsically unjust. I think you could still argue that it's necessary, and that need justifies unjust actions, but then we're back to wondering how exactly this plan is meant to actually work without some form of mass tyranny.
Of course, you would probably have a one child policy for those who are not sterilized.
I think you'd need dozens of pages of rules to cover all the loopholes, for example the rich paying off poor people to give up their reproductive ticket and carry the wealthy parents' additional offspring.
But that's by the by. If you can enact a global one child policy, there'd be no need whatsoever for mass obligatory infant sterilization anyway. If everyone in the world were to, from today, only parent 1 child then the population would drop precipitously, halving it within a couple of generations. However, if we can't convince the world's population to have only 1 child, what's the chance of convincing them to allow the state to draw lottery on whether their 1 offspring gets to reproduce?
I envision a sustainable World population of only one or two billion. How else can we possibly get there from here without violence?
Well, let's be clear: if you're prepared to commit violence to oblige people to conform to your idea, then you're bordering on a kind of belief that is typically reserved for the religious, so the chance of me being able to say anything that changes your mind is minimal. However, I can of course point out that I've already addressed your false dichotomy before, and explained that there are numerous other ways - vastly more rational ways at that - to change population demographics in order to preserve the environment.
Of course though, there's a damn sight more to it than just population. Consumption patterns are also vital to take into account here. At present rate, to get the same outcome with your sterilization lottery, you'd probably need to sterilize about 6 to 8 Africans compared to 1 American with respect to how many resources they consume. It wouldn't be much good if we killed off 6 billion people only to find that the remaining 1 billion were all selfish, greedy arsehats who felt the need to own 18 cars, 12 bedroom houses with swimming pools etc - couldn't we just have kept the 6 billion people whose consumption levels weren't anywhere near as threatening to the environment?
But I suppose we're circling back to the dysfunctional notion contended of 'fairness' there, where we should really be looking at real world solutions.