1919, American English, originally a reference to rumors of quadriplegics as a result of catastrophic wounds suffered in World War I (the military vehemently denied there were any such in its hospitals), from basket, n. + case, n.2. Probably literal, i.e., stuck in a basket, but basket had colloquial connotations of poverty (begging) and helplessness long before this. Figurative sense of "person emotionally unable to cope" is from 1921.