I answer you categorically yes! It is necessary that the mankind has to continue at all because we are the only ones -as far as we know- who possess those that we call intelligence. There are many reasons to affirm what I say. We can, in fact, find a reason for each branch of knowledge if we search. That's don't mean that the reason you asked is gnoseological, but our action of think on things and act according to our observation gives us a superior grade of dignity than all others known entities in the Universe. This is intelligence is dignity. We are all agreed that human beings has more dignity than other phenomenological entities, but that dignitive superiority of Mankind only can be based on our inteligence because there are no other differences between us and gorillas or other biological beings. So the development of our intellectual act is what gives us reasons to save the Mankind from the inexistence or the dissapearance and it justifies our wish of continuity is not related only to our emotional needs because emotion is a small, non-representative, fraction of the totality of the inteligence. Inteligence consists -in addition to other things- in the knowledge of reality that is, thus we can find reasons to our existence and continuity in the depth of the reality itself through our wisdom. I will give you a practical example of how do it the manner I propose, that is, justify our continuity in our inteligence. I'll show the most intrincate argument that I consider now, the rest are suchlike and not much more difficult.
The necessity of the continuity of de Mankind by its necessity of continuity
This is the more categorical argument I have found. For the concept of necessity of continuity we need the phenomenological notion of necessity, which is perceived by the human beings and provoked by a noumenological entity, but the necessity itself is something phenomenological -at least what we perceive-. The phenomenon of necessity are captured by human understanding through the intelectual act of perceive the forms of reality adapted to our senses in the experience which we received from the reality via our perception. Thereby if our continuity were contingent to Universe, the concept itself of necessity would be contingent and therefore there aren't necessary entities in the Universe except the concept of contingency, which would be the unique possible concept relative to existenciality with meaning mediated by our thinking. Hence the necessity of beings who postulate the necessity itself, i.e., human beings, is proved by the absolute necessity. Thus our categorical argument prove the necessity of continuity of mankind downright.