They actually denounced his positions very actively, which was unusual for the time. Soviet newspapers attacked him as a racist, fascist, and "advocate of nuclear war." He was a great figure to attack, of course, from their point of view, but they did seem to be genuinely unnerved by him. Pravda published an editorial cartoon of Goldwater as a plane shooting bombs out of his mouth (yes, I looked that up just for you after reading a description of it); that was before he ran, but you get the picture. When he lost, they issued both congratulations to Johnson, and also announced that Goldwater's loss was a referendum on strong anti-Soviet policies. Their conspicuous opposition to one candidate in particular, and their palpable relief when he lost, was noted in the American press at the time as unusual for Soviet coverage (which was usually more equal-opportunity unhappy with American presidential candidates).

This is only what was publicly published and announced by them; I do not have any sources that indicate more subtle internal negotiations, much less "public opinion" (which is hard to gauge in the Soviet state).