Here's what the OED online says about it:
quote:
basket case n. slang (orig. U.S.)  (a) a person (esp. a soldier) who has lost all four limbs;  (b) transf. one who is emotionally or mentally unable to cope; something that is no longer functional, esp. a country that is unable to pay its debts or to feed its people.
1919    U.S. Official Bull. (U.S. Comm. on Public Information) 28 Mar. 1/1   The Surgeon General of the Army‥denies‥that there is any foundation for the stories that have been circulated‥of the existence of ‘basket cases’ in our hospitals.
1944    Yank 12 May 17   Maj. Gen. Norman T. Kirk, Surgeon General, says there is nothing to rumors of so-called ‘basket cases’— cases of men with both legs and both arms amputated.
1967    Saturday Rev. (U.S.) 25 Mar. 30/3   Kwame Nkrumah should not be written off as a political basket case.
1972    Observer 24 Sept. 36/6   The ‘hero’, a legless, armless, faceless 1914–18 basket-case.
1973    Observer 15 Apr. 6/2   The real basket cases of European agriculture are the Italians and the Bavarians.
1978    S. Brill Teamsters vi. 227   He was a basket case because of Spilotro. A totally broken man, crying and whimpering.
1978    Puzo Fools Die xxiv. 275   ‘Hunchbacks are not as good as anybody else?’ I asked.‥ ‘No‥nor are people with one eye, basket cases and‥chickenshit guys.’
1982    Newsweek 11 Jan. 21/2   On a continent that is full of economic basket cases, the small, landlocked nation is virtually debt free.    
 After composing this I decided to do a Google search for the 1972 quote. I was surprised that the first hit was  
Wordcraft. The second was  
WorldCat. WorldCat describes 
Johnny got his gun, a 1938 book by 
Dalton Trumbo. WorldCat describes the book as a "...story of an American youth who survives World War I as an armless, legless, and faceless basket case with his mind intact." The book was subsequently serialized on radio, made into a movie, and turned into a stage production. 
The two sources that Kalleh cited, 
World Wide Words, by Michael Quinion, and 
The Phrase Finder, as well as the OED (all U.K. sources) say the origin of the term is U.S. 
The Word Detective (a U.S. source) agrees. But for some inexplicable reason,  
Wikipedia says it originated in Britain.