To be a communist party member or unenrolled supporter in that era means that one was willing to be a spy. I am hard put to imagine that Oppenheimer never volunteered nor was ever called upon to spy for the Soviets.
I have not read enough on Oppenheimer to be confident of an opinion. The movie is said to be a reasonably accurate adaptation of the source material, American Prometheus, which I have not read. The movie does show Oppenheimer being approached, more than once, but suggests that he deflected the approaches — while maintaining his ties to the people who had approached him and lying about the particulars later to the FBI.
At the very least, he was monumentally indifferent to security concerns, partly due to personal arrogance and partly because he was a man of the left, comfortable swimming in a red sea. To Oppenheimer, Hitler’s anti-semitism put Germany clearly in the enemy camp. I don’t know that he ever saw Stalin, the USSR, and communism as enemies — which was the default fellow traveler position. The anti-communist liberals of that era were honorable people who despised both Hitler and Stalin. But the “no enemies to the left” types tended to turn a blind eye to communism, and they provided useful cover for the spies and agents of influence.