I. DEGROWTH WOULD CURB EMISSIONS
The theory here is that some people, with a guaranteed income, would simply drop out of the workforce; that the labor supply contraction would lead to a drop in GDP; and that this reduction in economic activity would in turn lead to a decline in greenhouse emissions. This is probably the most popular take on this that I've seen, though I think that on these terms it ends up being the weakest.
One immediate problem with this argument, as given, is that it's not clear that UBI would necessarily shrink the labor supply. People work because they need income, but since they also work for other reasons, one can't simply assume that a UBI would catalyze a net reduction in work. If it turns out that UBI gives us the same "amount" of economic productivity (whatever that means) but simply changes the incentives, then an environmentalist argument for UBI that relies on generalized degrowth falls on its face.
A second problem is that even if UBI curbs economic productivity, it may not necessarily follow that this would curb greenhouse emissions. That's because emissions may not be a function of GDP; in the last few years, for example, the two have arguably uncoupled, and there are historical examples of countries that have seen simultaneous economic contraction and growth in carbon emissions. While most data seems to point to a relationship between the two, I don't think we can make assumptions about what would happen given a massive intervention aimed at global reductions in economic activity.
These objections may seem distinct, but I think that they are actually related, and if we bear them in mind we can build a stronger case:
II. EMANCIPATED LABOR WOULD BE GREEN
The degrowth argument, we have seen, rests on two basic assumptions: that people will only work if they need income, and that work necessarily produces greenhouse emissions. Both of these claims, I've suggested, may be faulty. But what if, instead, we suppose that wage-slavery disproportionately incentivizes certain kinds of work - in particular, work tied to the exploitation of fossil fuels?
If this is true, then UBI may very well free people up to engage in labor that produces fewer carbon emissions, with a net positive impact for climate change. This is an argument that can be made without generalized claims about GDP and "economic productivity" and how these things relate to UBI and climate change.
Here, what I think you would have to argue is that there is a direct relationship between wage slavery and the irresponsible use of fossil fuels. Conceptually, this strikes me as a pretty intuitive point: it makes sense that the same economic regime that recklessly exploits labor would also recklessly exploit fossil fuels. All you have to do here is make the standard Marxist environmental case that wage-slavery is what facilitates the bourgeoisie's mobilization of the means of production in an infinitely escalation pursuit of capital - and that technologically, those means of production have always relied in one way or another on burning fossil fuels.
Capitalists, in fact, implicitly recognize this point when they insist that the fix for climate change must be technological - that we cannot hope to place limits on economic activity, and must instead hope for some kind of technological silver bullet that lets us chase profits without burning coal. This point is certainly true if you take wage-slavery for granted; in that case, all you can do is dream of some magical form of production where infinitely expanding resource extraction somehow doesn't have deliterious environmental consequences. It seems to me, however, that no matter how politically and logistically difficult it is to launch UBI, that this approach is still more plausible than the probably-physically-impossible fantasy of infinite clean energy.
Regardless, what seems clear to me is that the fight for UBI and against climate change are both caught up in a basic political fight against capitalism. And for that reason, I think there's a third argument that's even stronger than the first two:
III. UBI WOULD INSTITUTIONALIZE DEMOCRATIC REDISTRIBUTION
You absolutely cannot fight climate change without massive redistribution. And the only way that you get there is by creating powerful institutions through which people can democratically expropriate the commonwealth from rich people who are hoarding it.
The problem is that this is a huge political lift, and since climate change is a progress trap, most people don't feel motivated to fight it. The consequences for failure, as apocalyptic as they are, are also extremely long-term, and by the time anyone feels motivated to do anything about carbon emissions it will probably already be too late.
But this, I think, creates an extremely strong case for UBI - because UBI creates a redistributive institution powerful enough to fight climate change that would provide immediate and significant material benefits for everyone. Once you enshrine the principle that everyone deserves a basic standard of living regardless of ideas about desert, that redistribution on the scale of (say) at least 10% of the national income is warranted, and that we should build a state institution powerful enough to guarantee this - from here, the case for funding green international development is open-and-shut. And more importantly, the systematic leverage that the rich have to fight it has been severely undercut.
Ultimately, I'm not sure how the fight against climate change ever gets off the ground without something resembling this approach. It's not clear that UBI would spark degrowth, or that degrowth would bring down greenhouse emissions; it seems possible that UBI would free people to engage in labor that's less environmentally destructive, but even this is fairly speculative. What strikes me as certain, however, is that people aren't going to fight for adequate redistributive institutions unless there is some kind of powerful, immediate benefit - something more than just the intellectual conviction about the levels of CO2 in the atmosphere. UBI is exactly the sort of incentive the fight against climate change needs.