Not having children because the climate will get worse is wrong, if you're in the First World: as ASX points out, conditions are so much better than in the Third World, and will stay that way; so if you genuinely believed the idea, you'd have to believe in no-one in the Third World having children.
Not having children because they'll emit CO2 is I think wrong, too. After all, we're expecting to get to carbon neutral at some point, like 2050 perhaps, so they're free after that.
And then some notes on some specific points.
* The current scientific consensus, as per leading scientific organizations like the United Nations’ Intergovernmental Panel on Climate Change, is that climate change will be very bad, but not world-endingly bad. And the link is to a Vox article Is climate change an “existential threat” — or just a catastrophic one? Vox isn't really an RS for this stuff; the reason that ACX doesn't link to the IPCC itself is of course because the IPCC says no such thing.
* Climate change will cause worse hurricanes, fires, and other disasters. It will lead to increased spread of invasive species and diseases. It will hit subsistence farmers in poor agricultural countries very hard, and some of them will starve or become refugees. But it won’t cause the collapse of civilization. It won’t kill everyone. Life in the First World will continue, with worse weather and maybe a weaker economy, but more or less the same as always. This is all a bit funky. Hurricanes, well, there are plenty of people to tell you about that. Invasive species are I think rather more spread by global transport and travel, not GW. The best hope for subsistence farmers in poor countries is that they stop being subsistence farmers, and that their countries stop being poor. Refer to Adam Smith for a short - but, it would seem, quite hard to achieve - set of conditions. The collapse of civilisation seems vanishingly unlikely. Life in the FW will not continue "more or less the same as always" because, errm, things will change; technology and politics and society will progress. GW will probably decrease GDP from the value it would have had with no GW, but GDP will continue to increase anyway, just by a bit less.
* focus on sea level rise because it’s easy to quantify and display. People often choose SLR because it is unambiguously bad; but it is hard to get any significant damages by 2100 because the expected rise just isn't big enough.
* let’s say there’s still a 1% chance that everyone’s wrong and [a runaway greenhouse effect] can happen. 1% is clearly not exact; it's a proxy for "unlikely, but not negligible". But this is wrong; the real assessment is more like virtually impossible, though as with all these things you never can tell. However, using a lower probability wouldn't affect his argument.
* What we actually need is concerted government action... But your choice not to have children makes that government action less likely to happen. Suppose 1-2% of Democrats stop having children because they’re worried about climate change. Meanwhile, Republicans don’t care about this and have just as many children as ever. Since children tend to share their parents’ political beliefs, this skews elections in favor of the Republicans, who will prevent strong government action. I don't think this makes any sense. Because politics simply doesn't work that way, with fixed party boundaries; instead, the parties shift to pick up voters.
Update: Chilling Effects
See-also his In what sense do 10% of people die of the cold? And why is heat-related death most common in Greenland? Note the Epistemic status: Extremely confused! Low confidence in all of this. But also the I’m not really impressed with the people working in this field.
* Authoritarian Left, Authoritarian Right by Pierre Lemieux
* CPI Bias vs. the Penn Effect by Bryan Caplan
* Highlights from the Nobel Committee Report by David Henderson, which I mention so I'll be able to find his criticism of the Card/Krueger minimum wage stuff that everyone is so keen to spout.