Today, we use basket case to mean someone who is crazy, or psychologically hopeless. It turns out the actual definition can be traced back to World War I. The phrase first came along as a slang term during World War I. It referred to quadruple amputees, or those who had lost all four limbs.
The reason for this is that soldiers who lost all their limbs were carried away in baskets. This wasn't because of a shortage of stretchers, but because if they had been carried on stretchers, they would have been too prone to falling out.
To be fair, this wasn't a very common occurrence. But since then, the definitions have broadened to refer to anything whose function is impaired. Still, the most common usage is in mental illness, specifically in cases where someone is incapable of behaving naturally.
Basket case has also referred to a country that can't pay its debts, an abandoned vehicle stripped of its parts, or simply an eccentric person.
Though the term was never exactly politically correct, thanks to the euphemism treadmill we mentioned in an earlier article (http://www.omg-facts.com/Language/What-we-consider-politically-correct-tod/51508), calling a quadruple amputee a basket case is today pretty rude.