Notice a key word missing here. They say that humans are affecting the Earth's climate. They do not say humans are SIGNIFICANTLY affecting the Earth's climate.
The whole thrust of Monckton's article is not that there is no effect, it's that they are grossly overestimating the effect
Further, the term 'significant' itself needs clarification. Suppose, for example, I have a card playing program. After running it, I make a small change to the seed for the random number generator. Have I "significantly" changed it?
On the one hand, I've totally changed the sequence of deals the program will produce. On the other hand, unless there was some defect in the generator, my change probably won't have affected the aggregate statistical properties of hands dealt.
I would suggest that even if humans are significantly changing the climate, they are not doing so in such a fashion as to have any predictable effect. It is probably true that had there not been so much CO2 in the atmosphere, some places that received rain today would not have, and vice versa. But so what? That a particular action may have random effects on already-random behavior shouldn't be any cause for concern. The behavior is still neither more nor less random than it was before.