Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) treats the expressions space-case, space cadet, and space out in a single entry:

space cadet or space-case or space-out n phr or n by 1980s A mad or eccentric person, esp one who seems stuperous or out of touch with reality as if intoxicated by narcotics: =NUT, SPACED-OUT: Alda presents her as such a space cadet that the agony of divorce is tempered—Kings Courier/ ...meant to convince the jury that he is an unreliable spaced-out, that perhaps he was hallucinating—Washington Post {probably fr the 1950s TV program Tom Corbett, Space Cadet, which followed the adventures of a group of teenage cadets at a 24th-century space academy, thought of humorously as being far out, way out, etc.}

Chapman & Kipfer notes that spaced-out meaning stuperous, intoxicated, crazy, or eccentric was in use by 1968. Although the link to an article in Social Problems, volume 24 (1976) (cited in phenry's answer) clearly identifies that instance of "space case" as street lingo, the term seems to have caught on fairly quickly among college students, as well.

Here are the three earliest matches for the term space case (and its variants) that an Elephind search turns up. From the "extraneous misclassifieds" column in the [Houston, Texas] Rice [University] Thresher (October 10, 1978):

Space Case, Had fun with our 'L' configuration brainchild. Remember our promise. Your 'roommate' IRBS

From "Learning This Jargon Can Be a Breeze," a United Press International story published in the Columbia Missourian (January 18, 1981):

College students have a jargon all their own. But the terms change with the times.

...

Here is a glossary of some other campus terms that may help parents understand their offspring and students at one campus understand those at another.

...

Space — Describes a person who is crazy, as in Space Case' or He's spaced.'

Space Out 1 To spread things out over a period of time 2 To forget something

And from "Lockheed Workers Claim Brain Damage from Chemical Reaction," in Synapse (the University of California San Francisco student newspaper) (March 5, 1981):

Metherell and Froidevaux, however, are showing more than symptoms: they aren't the men they used to be.

"He was a real bad case, it was scary bad," said an acquaintance who saw Froidevaux a month after the accident. "He wasn't the same person I knew before. I couldn't talk to the guy, he was kind of a space case."

So the details here and in phenry's answer support this summary of space case: "U.S. street slang by 1976, U.S. student slang by 1978."

Unfortunately, databases of newspapers and other periodicals from 1925 onward are excluded from the massive Library of Congress Chronicling America database, and the collections of more recent material that permit free public access are rather skimpy. A person who had access to collections not available for free might well find additional instances space case in the relevant sense from earlier in the 1970s and possibly into the late 1960s.

Update (September 18, 2020: Slightly earlier newspaper instances of 'space case'

A recent review of the Elephind newspaper database turns up three relevant examples of "space case" that are earlier than the October 1978 Rice Thresher example that my original answer cites.

From Fred Williams, "Men's Tennis Team To Begin Spring Season," in the [Poughkeepsie, New York] Miscellany News (April 8, 1977):

This year's batch of quasi-athletic pseudo-intellectuals comes in a various assortment of shapes and sizes, some with their grey matter in the pink, others whose grey matter is on a LOA. Cliff "Just work and play tennis" Bereck has been playing the No. 1 single spot, while Fred Williams and Mike Reese have been out of the running, Mike "Kansas City Space Case" Reese is recovering from major brain surgery while Fred "lost the keys" Williams has been in the market for a new knee and somebody to tell him not to play football.

From an item on NFL football in the San Bernardino [California] Sun (November 15, 1977):

Miami's space case

MIAMI (AP) Veteran linebacker Bob Matheson calls A.J. Duhe, the Miami Dolphins No. 1 draft pick, a "space cadet."

"AJ.'s way up here," said Matheson, holding his hand above his head. "A loosey-goosey guy ... if there's such a thing as a cool guy, Duhe's cool."

And from "Stage: For Colored Girls," in the [San Francisco, California] Bay Area Reporter (August 17, 1978):

Sometimes pretty, sometimes painful, often funny, always soul-searing, the event sent the overwhelmed standing ovation audience into the streets a collective space case.

So we have a Vassar College tennis player with the nickname "Kansas City Space Case" in April 1977; a California newspaper headline equating "space case" with"space cadet" in November 1977; and an audience that had just been transported by an effective stage play characterized as "a collective space case" in August 1978.