Without the mass extinction of the dinosaurs, is doubtful that human beings would ever have evolved in the first place. By comparison to the effects of the that meteorite, all of the human activity in the world that has ever occurred since the first human being who hunted for prey or started a fire is infinitesimally small and utterly insignificant.
More importantly, human concern for animal species extinctions seems to be largely predicated on our anthropomorphic impulses: that is, we have the greatest empathy for animals that remind us of ourselves or that seem appealing or "cute" to us. Consider the different way that we regard tuna and dolphin for just one example. We hunt the former so aggressively that we are on the verge of having to maintain wild tuna populations artificially if we hope to continue eating as much sushi and tuna fish sandwiches as we wish. Other than our self-interest in having enough tuna to eat, we express little if any empathy for the millions of tuna that are herded into nets, hooked, and clubbed to death. What (some) humans do to sharks is even worse: Japan, in particular, routinely fishes for shark and because the only economically valuable part of the shark are their tails and fins, they simply haul them out of the water, slice off their fins, and toss them back into the ocean to suffocate and drown as they sink to the bottom of the ocean.
Meanwhile, every can of tuna fish now boasts a symbol indicating that no methods of fishing for tuna were used that might have endangered dolphin in the process. We value dolphin because their intelligence and social behavior mirrors our own more closely and because they interact with us instead if eating us when given the chance to do one or the other. More generally, this is the way many who express concern over species extinctions choose the objects of their concern. Objectively, there is nothing more or less "tragic" about the extinction of a species of dolphin than of a species of shark or tuna. For that matter, when was the last time anybody ever expressed concern or raised donations to preserve any of the hundreds of thousands of species of beetles that might be on the verge of extinction?
In principle, extinction is nothing more or less than a necessary element of biological evolution. On one hand, there is nothing necessarily "wrong" with artificially preserving specific species that benefit human life; on the other hand, there is nothing tragic about the extinction of any animal species. More importantly, to the extent the extinction of species is a good or bad thing, it cannot be that the determination of the value of a species is whether it reminds us of ourselves or provokes a nurturing response in us because it happens to have large eyes and fluffy fur that make it "cute"…