“The new world will be a place of answers and no questions, because the only questions left will be answered by computers, because only computers will know what to ask.”

Source: Burnt Offerings

Source: Fourth Realm Trilogy (2005-2009), The Traveler (2005), Ch. 2

Homecoming saga, The Memory Of Earth (1992)

From "President Addresses the Nation in Prime Time Press Conference" http://georgewbush-whitehouse.archives.gov/news/releases/2004/04/20040413-20.html, Washington, D.C., on why the President and the Vice President insisted on appearing together before the 9/11 Commission, rather than separately. (April 13, 2004)
2000s, 2004

Quoted by Max Weber in his lecture "Science as a Vocation"; in Lynda Walsh (2013), Scientists as Prophets: A Rhetorical Genealogy (2013), Oxford University Press, p. 90

[Undecidability and intractability in theoretical physics, Physical Review Letters, 54, 8, 1985, 735–738, 10.1103/PhysRevLett.54.735, https://www.stephenwolfram.com/publications/academic/undecidability-intractability-theoretical-physics.pdf]

As discussed in this entry from Quote Investigator http://quoteinvestigator.com/2011/11/05/computers-useless/#more-2932, the origin seems to be the article "Pablo Picasso: A Composite Interview" by William Fifield which appeared in The Paris Review 32, Summer-Fall 1964, and collected a number of interviews Fifield had done with Picasso.
Common later variant: "Computers are useless. They can only give you answers." This variant seems to have arisen in the 1980s, the earliest known appearance in a book is Herman Feshbach, "Reflections on the Microprocessor Revolution: A Physicist's Viewpoint", in Man and Technology (1983), ed. Bruce M. Adkins, where the attribution is described as "rumoured". http://books.google.com/books?id=9EohAQAAIAAJ&q=Picasso
1960s

Variant: People only ask questions when they're ready to hear the answers.
Source: The Cider House Rules