Has the earth benefited from the presence of the human species? What is the method to determine a blessing or a curse that relates to human existence?
There are two possible answers to your question:
Without the existence of humans, the question dissolves. The very concepts of "good", "bad", "blessing", and "curse" are based on human thoughts, language and self-awareness, and don't have any meaning independent of the human subject. So the statement "the earth would be better off if humans didn't exist" doesn't have any meaning.
The second answer is similar to one I gave to another question: It is possible that humans are the only self-aware agents on earth, and as such are the only way by which the earth itself becomes self-aware. This would imbue humans with a special value that other living organisms don't have. From this point of view, one could answer that the earth is better off because of human existence, no matter what other damage and tragedies humans cause. This assumes a self-aware earth is better than a non-self aware one. See this question as well for a related discussion.
While I agree the question is pretty faulty, there might be some ways to reasonably approach it.
One needs a value system of some sort that is somehow measurable and arguably applies beyond the actual "human beings" who will be doing all the defining and measuring. This would require some sort of second-order, reflexive, or transcendental argument. We draw as close as "possible" to the singularity of the notorious "god's-eye-view," from which we abstract or bracket purely "human" interests. (In addition to theology and nihilism, there are various attempts in so-called speculative realism to present such conditionally "inhuman" thinking.)
Candidates for extrapolated values might include biomass or sheer "quantity of life" or "diversity of life" or, more common, "reduction of suffering" or "acceleration of genealogical speed" or "interplanetary communications of lifeforms" or "time reversal" whatever. I am sticking with "life" here since it at least has some sort of directionality vis-a-vis entropy. These could then plausibly act as coordinate values within which "human life as we know it" is an independent variable.
In fact, people do already think in terms of biodiversity and "total suffering" among sentient beings and so on. So some inchoate seeds of such "inhuman" values systems exist. And we have value systems in which both "life itself" and God or "Afterlife" or perhaps the Ubermensch are "more valuable" than mere "extant humanity" per se, which may disappear while the "value conditions" remain "post-human." Which is assumed delusional from our present physicalist dogmas.
The problem is that conceding a minimal "value system" requires only some coordinates and a direction or "attractor," so dropping the "human" requirement gives us infinitely many, all seemingly meaningless. How to select? Most humans will undoubtedly vote in our own species "fruitful" self-interest, though something in us is strangely capable of imagining otherwise. There are certain very appalling moments in human history when I myself might be willing to let biomass prevail and plankton inherit the earth.
In my opinion the question does not seem meaningful. There is no criterium to judge what is good or bad for the earth.
The terms good and bad are meaningful with respect to individual animate beeings like animals and humans. Moreover, one can discuss whether this concept applies to intelligent artificial systems too. But the concept does not apply to inanimate objects like the earth.
A different question would be: Is the human species good or bad for the animals of others species? But also here one first needs a consensus about the meaning of these concepts.