An infirm or failing person or thing - unable to function properly. Originally this referred to soldiers who had lost arms and legs and had to be carried by others. More recently it has been used to denounce a failing organisation or scheme and is less often applied to people.
In its original meaning this term comes from the US military immediately following WWI. Strangely, it was never used to describe an actual person but only in denial of any such servicemen existing. This bulletin was issued by the U.S. Command on Public Information in March 1919, on behalf of Major General M. W. Ireland, the U.S. Surgeon General:
"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."
This bulletin was reported on in many U.S. newspapers at the time. Many of them also defined what was meant by 'basket case'; for example, this from the New York paper The Syracuse Herald, March 1919:
"By 'basket case' is meant a soldier who has lost both arms and legs and therefore must be carried in a basket.
Given that the term was originally reserved for incapacitated servicemen, there wasn't much call for it until the next major war of English-speaking peoples - WWII. Again, it comes from the U.S. military and again in the form of a denial from the Surgeon General. In May 1944, in Yank, the Army Weekly, the then Surgeon General, Major General Norman T. Kirk, said:
"... there is nothing to rumors of so-called 'basket cases' - cases of men with both legs and both arms amputated."
Clearly, given the scale of the casualties in both wars, there must have been cases of multiple amputation. It isn't recorded what term the U.S. Surgeon General used to describe these - clearly not 'basket case'.