Thinking narrowly and parochially, and in a short time frame, as humans are wont to do, it may be tempting to think that humans must somehow preserve the current state of the Earth exactly as it is; as if it's your aquarium with tropical fish. The reality is that the Earth changes, and so do biological organisms, humans included, of course.
When we stop thinking within such a small scale in term of space and time, then Hawking is completely correct. It's a matter of perspective. Humans are not used to long term planning; that is, time scales larger than a generation or two. But, despite our shortsightedness, the future nevertheless happens.
It might be helpful to review the Kardashev scale (http://en.wikipedia.org/wiki/Kardashev_scale) which considers the evolution of intelligent entities on a longer time frame than we are used to thinking. It was conceived by Russian astronomer Nikolai Kardashev to categorize the levels that an intelligent civilization might reach, in terms of it's energy usage. On his scale, we are not even to level one yet. That points out another canard which humans tend to enjoy, which is that they consider themselves advanced, intelligent, and powerful. Maybe someday junior, but don't get ahead of yourself.
Physicist Freeman Dyson's concept of the next step in the evolution of how we and other intelligent entities elsewhere in the univese might live in space after we outgrow the home planet on which we evolve (http://en.wikipedia.org/wiki/Dyson_sphere) is also worth considering. Our home planet feels very comfortable and safe, just like our bedroom, blanket, and teddy bear when we were children, but we have to grow up. The Earth is just one of millions of planets; we should be careful to not be so sentimental about it when considering our options; we all stop living with our parents eventually.
Finally, it's also worth considering that not only will the natural biological evolution of the human species continue, but we are evolving and will continue to evolve in non-biological ways. We now consider the wearing of glasses to be normal. Soon, we will accept a computer implanted in our brains as normal. And, someday our entire self, that is, our mind, may be moved into an artificial substrate; ie. a computer.
When that happens (not if), then it will "feel" totally normal to say good bye to our security blanket; the Earth.