Who first said "We can predict everything, except the future"?
Solution 1:
A preliminary digression
Although I want to provide a useful answer to the poster's specific question, I must first point out the inaccuracy of Dougvj's answer. According to that answer, the phrase "We can predict everything, except the future" may originate in a longer block of observations about the future—
The future isn't what it used to be. We can predict everything, except the future. The future's uncertain and the end is always near. There is very little future in being right when your boss is wrong. If you do not think about the future, you cannot have one.
—and that entire block of text may have been written by John Galsworthy. In fact, however, the block quote is a pastiche of jokes, proverbial wisdom, and quotations of miscellaneous provenance, ending with a line that is indeed drawn from Galsworthy's Swan Song (1928):
Mr. Montross shook his head. "I know them, you see, Mr. Mont; if these people thought about the future, they could not go on living. And if you do not think about the future, you cannot have one."
The source of the saying 'We can predict everything, except the future'
As for the quotation that the poster asks about, the earliest instance of that approximate wording that I could find in a Google Books search is from David Redburn, "The 'Graying' of the World's Population," in Social Gerontology (1998):
An oft-stated demographer's joke comments, "it is easy to predict everything except the future" and while this is demographic humor, or lack of it, it does relate the trepidation with which population specialists approach projections.
Charles Doyle, Wolfgang Mieder & Fred Shapiro, The [Yale] Dictionary of Modern Quotations (2012) identifies a kindred expression as a "modern proverb":
Never make predictions (It is difficult to prophesy, It is dangerous to prophesy), especially about the future.
1956 "Proceedings of the Meeting," Journal of the Royal Statistical Society (series A, General) 119: 147 (Bradford Hill speaks): "Alas, it is always dangerous to prophesy, particularly, as the Danish proverb says, about the future." 2004 Toledo {OH} Blade 15 Sep.: "An old adage holds, "Never make predictions, especially about the future.'" Y[ale] B[ook of] Q[uotations] Bohr (2). The proverb is frequently referred to as Danish (or Chinese or Romanian) or attributed to a particular Dane (often the physicist Neils Bohr) or to Yogi Berra, Casey Stengel, Samuel Goldwyn, or Mark Twain.
Fred Shapiro, The Yale Book of Quotations (2006) does indeed attribute a version of the quotation to Niels Bohr, albeit with a fair bit of hedging:
It is difficult to predict, especially the future. Attributed [to Bohr] in Mark Kac, "Statistics" (1975). Kac states that this saying may have been "an old Danish proverb." K. K. Steincke, Goodbye and Thanks (1948), quotes it as a pun used in the Danish parliament in the late 1930s.
The earliest instance of "never make predictions, especially about the future" that a Google Books search turns up is from Ninth Circuit News (1979) [combined snippets]:
It was said of Trotsky that it was proof of his farsightedness that none of his predictions had yet come true. And Samuel Goldwyn said: never make predictions, especially about the future. Though I now qualify as an expert, having moved more than five hundred miles from home, I'll resist the temptation to make predictions.
Likewise, an address by New Zealand Prime Minister R.D. Muldoon to the London Chamber of Commerce (May 16, 1983) in the New Zealand Foreign Affairs Review (1983) [combined snippets] pins the tail on Samuel Goldwyn (albeit in somewhat different language), this time with a further nod to Mao Tse-Tung:
Sam Goldwyn said: "Never prophesy, especially about the future." In a similar vein, Mao Tse-Tung said: "Those who forecast should forecast often." When we get people as diverse as Sam Goldwyn and Chairman Mao agreeing on the same point, perhaps we ought to listen.
But a class note by Hugh T. Kerr of Princeton, New Jersey, in Princeton Alumni Weekly (July 11, 1984) attributes the thought to Yogi Berra:
An executive committee of sorts met to plan future events, but mercifully adjourned in time for cocktails. As they say, when all is said and nothing done, the committee meeting is over. You'll be hearing about class plans in due course. As Yogi said. “I never make predictions, especially about the future."
And Michael Kramer, "Taxing Details," in New York Magazine (June 3, 1985) goes up the ladder and credits Berra's former manager of the New York Yankees, Casey Stengel:
Before you bet your deductions, keep in mind what Casey Stengel once said: "Never make predictions, especially about the future."
The earliest Google Books match for "it's dangerous to prophesy, especially about the future" is from Scripta Instituti Donneriani Aboensis (1967) and attributes it to "a Danish humorist" (although not necessarily Niels Bohr in his role as standup comedian at conventions of physicists):
Looking ahead toward the future. As a Danish humorist has said "It is always dangerous to prophesy,—especially about the future". Being no prophet, however, I can freely prophesy—nobody will be surprised if I am wrong.
For its part, Sucrochemistry: A Symposium Sponsored by the International Sugar Research Foundation and by the Division of Carbohydrate Chemistry at the 172nd meeting of the American Chemical Society, San Francisco, Calif., Aug. 31–Sept. 2, 1976 (1977) credits Oscar Wilde with having formulated the wording:
On this optimistic note, I would like to remind you of Oscar Wilde's apt phrase that, "It's dangerous to prophesy, especially about the future".
Yet another formulation appears in James White, Protestant Worship: Traditions in Transition (1989):
Being a prophet is always a dangerous profession because the only thing we know about the future is that it will be different from both past and present, and that may change!
And this, from Stephen Blaha, The Life Cycle of Civilizations (2002):
Needless to say the prediction of human events is a risky proposition so this author (and hopefully the reader) realizes that “the future is subject to change without notice.”
Conclusion
The problem with the quotation "We can predict everything, except the future" isn't that no one ever gets credit for having originated it (or something like it)—it's that it's that far too many people do. Depending on the form of the wording you pursue, you can find instances dating back more than 60 years, but no specific person stands out as having recorded the idea as an original thought: even the earliest expressions of the idea in print tend to attribute the expression to someone else or to proverbial usage.
This is hardly surprising. The notion that prognostication is a risky business is surely millennia old. As an unidentified speaker in Proceedings of the Annual Convention of the National Association of Animal Breeders, volume 11, Parts 1958-1962, puts it,
Only crystal ball gazers (who get fees for it) and fools dare to predict the future. And only a fool is certain.
However, I will go out on a limb and assert, without exhaustive evidence to corroborate my claim, that the originator of the expression was not John Galsworthy.