The Soviet Union was very supportive of the Zionist cause at first because there were a plenty of Socialists in the Zionist camp (vis all the kibbutzim). Stalin hoped they would immediately join the Socialist camp upon granting the (first after the United States) official recognition in 1948. He was wrong. What followed were antisemitic purges (Arrests of the Jewish Antifascist Comittee in 1948-49, Night of Murdered Poets in 1952, Doctor's Plot Trials 1952-53) and more antisemitism dressed into the usual Soviet anti-imperialist rhetoric which was spread from all the official media of the USSR until it's very demise.

However, you are very wrong to presume that only the Soviet post-1948 stance is represented on the political left and only the American post-1948 stance is represented on the political right. Both "the left" and "the right" are very diverse in their views on Israel and every nation has a very different political tradition. As a rule of the thumb demand for the euro acceptance of the new (Soviet) antisemitism soared when European mainstream parties had to explain a change of attitude towards the Jewish State after the oil crises in 1967 and 1973.

source: History of the State of Israel by Michael Shterenshis.

There are still enclaves within the American political left and right that feel the opposite of how you characterize them. With that being said, a key point in transition for how Americans as a whole viewed Israel came in the five year period between the Six Day War and the Yom Kippur War. At a time when American soldiers were fighting a seemingly unwinnable guerrilla war in Vietnam versus Soviet allies, Israel was winning total victories against Soviet client states. It solidified in US public opinion the perception of the tiny social democratic, western enclave holding its own against a monolothic anti-western force.

I recommend reading: The Holocaust and American Life by Peter Novick