1590s, "money box;" also "money in hand, coin," from Middle French caisse "money box" (16c.), from Provençal caissa or Italian cassa, from Latin capsa "box" (see case (n.2)); originally the money box, but by 18c. the secondary sense of the money in it became sole meaning. Cash-crop is attested from 1831; cash-flow from 1954; the mechanical cash-register "machine for automatically recording the sums of money deposited in it" is from 1878.
Like many financial terms in English (bankrupt, etc.), it has an Italian heritage. Not related to (but influencing the form of) the colonial British cash "Indian monetary system, Chinese coin, etc.," which is from Tamil kasu, Sanskrit karsha, Sinhalese kasi.