One selling point of basic income is that there's no loopholes and there's very little bureaucracy costs, therefore a larger portion of the money is actually used for consumption instead of doing unnecessary (for society) work.
If you create a significant CO2 tax separately from some specific program, like basic income, the fear is that it will be (and will seen to be) just a general supplement on government spending. Which has a tendency to grow until it cannot, so you're just putting off an inevitable crunch that we're already bumping up against.
But if you tie them together, the pain of the tax is balanced by the pleasure of the income. And people have been shown to enjoy that sort of thing. Witness the popularity of tax refunds, even though getting one is strictly worse for you than not getting one.
You can then offer tax credits for CO2 capturing (with the burden of proof on the claimant).