During the summer of 1981, when it looked like parts of New York and New Jersey were going to run short of water, many businesses and homes were affected by the stringent restrictions on the use of water.
(A) it looked like parts of New York and New Jersey were going to run "looked like" is used to express similar appearance. This sentence is trying to express a future possibility and not similarity in appearance. Eliminate.
(B) it looked as if parts of New York and New Jersey would have run Same error as in (A). Also, incorrect use to present perfect "have run" for a future possibility. Eliminate.
(C) it appeared that parts of New York and New Jersey would run Correct answer - above errors are rectified and no new errors are introduced.
(D) appearances were that parts of New York and New Jersey would run "appearances" is a plural noun which is incorrect in the present context. We want to express a possibility and not talk about literal "appearances" here. Eliminate.
(E) it was the appearance that parts of New York and New Jersey would be running As in (D), use of noun is incorrect. Also, use of present continuous "running" for a future prediction is incorrect. Eliminate.