The earliest citation I can find for cut a check is actually from Safire’s NYT ”On Language” column from 24 May 1981. Safire writes:
Recording engineers, accustomed to “cutting records” (from a lathe cutting a groove in lacquer), use the verb “to cut” metaphorically. Reports Mr. Howarth: “I will ask the comptroller to cut a check for the amount necessary.” Similarly, military and naval types will recall the cutting of orders, which probably stemmed from cutting a stencil.
That sense of “cut” could be applied to a deal: “I think it goes back to the graphic arts,” suggests Thomas Turley of Harrison, N.Y. “When copy was finally decided upon, it was cut into a metal plate either mechanically or by chemical action. The product of this effort was known as a ‘cut’ as was the process itself, ‘to cut a plate.’ In that sense, ‘cut a deal’ means reducing a contractual exchange to fixed terms, written or otherwise.”
My vote goes to coinage by the record industry, although I’m willing to sit with anyone and do an etymology based on other specific citations.
[I didn’t bother to type out Safire’s blathering about a possible origin relating to circumcision which he includes only in order to make a bad pun about “going the extra mohel."]
The OED records the recording industry usage of cut from 1937.
Unless significantly earlier citations for cut a check are found, my bet would be that it comes from the stencil usage and the military “cut an order.” Safire’s 1981 notice of the term points to a coinage for cut check some ten or twenty years earlier, which would be at the height of the cold war when there were many in professional life who had served in the military. That, plus influence from the recording industry, would have given “cut” a sense of to make.