1919, United states English, originally a mention of rumors of quadriplegics due to catastrophic wounds suffered in World War we (the military vehemently rejected there have been such in its hospitals), from basket (letter.) + situation (n.2). Probably literal, i.e., stuck in a basket, but basket had colloquial connotations of impoverishment (begging) and helplessness a long time before this. Figurative sense of "person emotionally struggling to cope" is from 1921.
If anything is a basket case, its so incredibly bad that it cannot be aided.
They said he'd been a basket case for almost two months by the time he jumped.