Many people on Less Wrong believe reducing existential risk is one of the most important causes. Most arguments to this effect point out the horrible consequences: everyone now living would die (or face something even worse). The situation becomes even worse if we also consider future generations. Such an argument, as spelt out in Nick Bostrom's latest paper on the topic, for instance, should strike many consequentialists as persuading. But of course, not everyone's a consequentialist, and on other approaches it's far from obvious that existential risk should come out on top. Might it be worth to spend some more time investigating arguments for existential risk reduction that don't presuppose consequentialism? Of course, "non-consequentialism" is a very diverse category, and I'd be surprised if there were a single argument that covered all its members.
This should depend very much on the system in question. Some people have already given examples of possible philosophical stances but it is worth nothing that even some religions can plausibly see existential risk as a problem. For example, in most forms of Judaism, saving lives is in general important and there's also an injunction that one cannot rely on divine intervention for any purposes. So even if someone believes that there's a nice deity who will make sure that existential risks don't occur, there's still an obligation to take active steps in that regard. This dates back to ideas which are directly in the Bible, such as the exchange between Mordechai and Esther in Chapter 4 of the eponymous book.
Similar remarks can probably be made at least for the other Abrahamic religions in various forms, although I don't have the knowledge base and time to flesh out the details.