In this Washington Post article, Norman Platnick of the American Museum of Natural History is quoted as saying the following:
If spiders disappeared, we would face famine.
Spiders are primary controllers of insects. Without spiders, all of our crops would be consumed by those pests.
I also found this article that says the following:
If we took spiders off the Earth, we would all die in less than 5 years.
Without all those hungry, carnivorous spiders, insect populations would explode, food crops would be decimated, ecological balances would be ravaged, and humans would probably starve within a matter of months – if they hadn’t already died from insect borne diseases.
The idea that humans would go extinct if spiders disappeared is bolstered by Platnick in an interview in which he questions if humans would even exist without spiders:
Norm: [...] And they are in fact the dominant predators of insects, and without them we'd be in dire straits. In many cases most of our crops, for example, would be totally destroyed by the insects that already do take a large toll on our crop production, but the spiders help control them.
Dr. Biology: [...] What if there were no spiders on Earth, what would the world be like? Would many of the other living things still be here? Would we even be here?
Norm: I'd say it's questionable whether we would be here.
Are spiders really that important to crops and humans?