You can picture something that no longer holds its own shape. It has come apart, so it cannot stand, walk or work by itself, and whatever is left has to be gathered up and carried in a basket by somebody else. Every use of this phrase points at that state: not damaged, but no longer holding together, and now dependent on others.
From the word’s history
The parts are basket + case. The source records a First World War rumour behind the phrase: soldiers hurt so badly that they had to be carried away in baskets. The same picture of a body that can no longer carry itself is what later got attached to people under stress and to countries that cannot keep themselves going.
這個詞由 basket（籃子）加 case（案例、病例）組成。辭源資料記下的來路是一次大戰時期的傳言：傷重到必須被抬著送走的士兵。後來這個「自己撐不住、要被抬著」的畫面，被移到壓力下的人身上，也被移到撐不住的國家身上。
Wiktionary, "basket case"
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