I know that as a result of the policy that atheism was to be the only public religion, the Soviet Union did their best to attack Christianity. However, I am interested in knowing the extent that practitioners of Judaism were persecuted.
It is a common misconception in the West to confuse Jews and Judaism, and as a consequence, antisemitism with attitude to Judaism. In Europe, especially in Eastern Europe, "Jew" does not mean religion, but means "ethnic origin" (whatever this means). Most people designated as Jews in Soviet Union were not religious.
The question is about religion (Judaism) so I will address the religion. Marxism in general is an atheistic world view, and Communists are against all religions. However the specific policies varied with time. In the early years of Soviet Union, no distinction was made between Christianity and Judaism. All religions were equally bad. And they were tolerated to some extent. But churches were closed, and destroyed, or closed and robbed, and synagogues were as well. There was no difference.
The change came during WWII, with the rise of (Soviet) antisemitism. (Antisemitism is a negative attitude to ETHNIC Jews, it has little to do with religion). One can describe the situation in this way: the attitude to Judaism became worse, while the attitude to Christianity better. Judaism was never outlawed, or prohibited (some religions WERE prohibited!), but teaching of Hebrew was. (The "official" language of Soviet Jews was Yiddish).
The situation became even worse when Soviet attempts to dominate the new state of Israel failed. A new wave of antisemitism started. It receded with Stalin's death and a new wave started during the 6-days war.
And these changes happened not because of any special intrinsic properties of Judaism, but for the sole reason that Judaism is usually practiced by Jews (in the ethnic sense). Ethnic Jews were discriminated (since WWII) and as a result, the attitude to Judaism was worse than the attitude to Christianity. The practice of Judaism was strongly discouraged but never prohibited.
(Some other religions were formally prohibited, like the Uniate (=Greek-Catholic). Even now this is the case in Russia. People go to prison in modern Russia just for being Jehovah vitnesses.)
Ethnic Jews were discriminated in hiring and education. But teaching Hebrew could qualify as "Zionist activity" and one could be imprisoned. Still, in big cities synagogues existed and functioned.
Depends on the era and circumstances
USSR was officially atheist, and all religion institutions were tightly monitored and controlled. However, every religion was not in same position, and level of persecution (or lack of) varied throughout the years.
In early years of USSR, many of the communist leaders were themselves Jewish. Antisemitism was strictly persecuted. Judaism itself was not officially protected, but it was in much better position than Russian orthodoxy (and Russian traditional culture in general). The later was seen as a "reactionary" force from previous regime (Empire) and was brutally suppressed. However, Judaism actually expanded in that period. For example, there were 13 synagogues in Leningrad (St Petersburg) in 1917, and 17 in 1927. 1920's were held as best years for Judaism in USSR.
In the years before the WW2 (Stalin's purges), Jewish communist leaders were gradually replaced (and sometimes killed in said purges), while state took somewhat lenient stance towards Russian Orthodox Church. Stalin's Socialism in One Country become ruling ideology, while those advocating world revolution and internationalism were removed. Jews in general had less power, but they still occupied mid-level party positions. Jewish Autonomus Oblast was formed at that time, with accompanying religious institutions. They were all state controlled, but there are no records of major persecution. Still, number of synagogues in major cities gradually dropped, as people were forced either to become secular or to move to JAO.
During the WW2, there was general feeling in USSR that Jews avoid military service, especially in front-line units and dangerous military branches (infantry, tank units etc...). It was often said that Jews serve on Tashkent front (Tashkent being in Asia and far from any danger) . Anyway, official Soviet policy at that time tried to protect Jewish civilian population as much it could, and persecution of religious institutions was almost completely abolished. They on their part gave support for war effort.
After the war, Zionist movement and newly founded Israel gravitated towards West. Soviets felt betrayed, and result was campaign against Rootless Cosmopolitanism. Nevertheless, it appears that official Judaism was not targeted. For example, in 1953 during Stalin's illness, service and prayers were held in Moscow Synagogue for his health.
During the era of Khrushchev, Stalin was regarded as too lenient on religion (contrary to popular belief, Khrushchev was actually less liberal in this regard ! ). There was large anti-religious campaign during his reign (1958–1964). It targeted mostly Russian Orthodox Church. Institutions of Judaism at that time were already very weak, so they needed no special attention.
After Khrushchev, during the final years of USSR, policy towards Jews was twofold. USSR was enemy of Israel and Zionist movement. Jews who supported this were either persecuted or simply allowed to immigrate. On the other hand, USSR tried to win over and assimilate most of the other Jews, especially those necessary for scientific and technical development. One example of this would be Gavriil Ilizarov, famous medical doctor of Jewish descent who won numerous Soviet awards. Official position towards Judaism remained essentially unchanged - it was closely controlled and kept away from Israeli influences.
All religious institutions in the USSR were treated as honeypots, and that, of course, was true about Jewish institutions doubly both because of traditional antisemitism and the fact that Judaism associates all Jews around the world was incompatible with the soviet "шпиономания" (obsession with espionage threat).
- Yevsektsiya was created in 1918 with a mandate of destruction of traditional Jewish life, the Zionist movement, and Hebrew culture.
- Teaching Hebrew was a felony "zionist propaganda"
- Opening a synagogue required 20 persons ("двадцатка") to apply, and all of them would immediately lose any hope for anything other than a menial job and be pressured to become KGB informers
- Shabbat observance was complicated by a 6-day school week and mandatory public education (religiously observant families with children had to constantly invent excuses and bribe officials to avoid persecution)
- Access to kosher food was limited: one had to know a shochet personally and bring a (live) chicken to him
- Wearing visible Jewish attire (kippah or tzitzit) was unthinkable
PS. See also my other answer.