The existence of alien life would validate their cherished belief that life can (and did) arise spontaneously by itself here on Earth. And if it happened here, then it must have also happened elsewhere in this seemingly infinite universe given the 14 billion years they currently consign as the age of the cosmos.
But what if we are alone in the cosmos and there is no alien life elsewhere? It is the death knell for their theory of origins and leaves only one other option that is devastating to their worldview - special creation by an omnipotent Creator. Thus, the stampede with ever increasing urgency to find alien life which in their minds would finally vindicate their position and prove the fallacy of a creator God.
In a forlorn bid to cover all the bases, some nautralists are cautiously advocating that if alien life does exist, it might be "extremely rare". See here.