coarse (adj.)

early 15c., cors "ordinary" (modern spelling is from late 16c.), probably adjectival use of noun cours (see course (n.)). Originally referring to rough cloth for ordinary wear, the sense of "rude, vulgar, unpolished" developed by c. 1500 and that of "obscene" by 1711.

Perhaps via the notion of "in regular or natural order," hence "common, vulgar" (compare the development of mean (adj.), also ornery from ordinary). Or it might be via the clothing sense, and the notion of "wanting fineness of texture or elegance of form." Or both, and there might be also an influence, via metathesis, of French gros (see gross (adj.)), which underwent a similar sense development. Related: Coarsely; coarseness.

Others are reading

Definitions of coarse from WordNet

coarse (adj.)
of textures that are rough to the touch or substances consisting of relatively large particles;
a coarse weave
coarse meal
coarse sand
Synonyms: harsh
coarse (adj.)
lacking refinement or cultivation or taste;
he had coarse manners but a first-rate mind
Synonyms: common / rough-cut / uncouth / vulgar
coarse (adj.)
of low or inferior quality or value; "of what coarse metal ye are molded"- Shakespeare;
Synonyms: common