20-Year Prospect
Humans have cared about animals since pretty much the dawn of time. Just look at Fred Flintstone (although he probably shouldn't use quite so many dinos in his dishwasher).
People will continue to care about animals for a long time. However, wildlife rescue as a career is a relatively new thing. It hasn't always been possible to get paid for pulling cats out of trees.
Despite the youth of the industry, its future is looking bright. The industry is expected to keep growing, and not at a snail's pace—we're talking 23-28% job growth between 2010 and 2020, depending on which area of the field you work in (source). There will be tons of new jobs in this field in the coming years.
This is great news for you, but keep in mind that the presence of more jobs won't eliminate the competitiveness of this field. Lots of people want to be paid to save animals. Give yourself the edge by earning a degree and some certifications and putting in some volunteer hours early in your career, and the next twenty years should be pretty rosy.