Modified from an earlier answer of mine

Although the USSR initially supported the creation of a Jewish state in Palestine in 1948, it soon reversed course and became one of the firm supporters of the Arab states and shrill condemnations of Israeli actions became one of the dominant leitmotifs inside the Eastern bloc press. Part of the reason for this reversal was certainly geopolitical. The decolonization process and the emerging Cold War meant that gaining the support of the Arab world offered potentially more advantages to Soviet foreign policy than a narrow strip along the Mediterranean. But bald geopolitical considerations was only part of the story, and these imperatives coexisted along preexisting antipathies within the Soviet state both towards Zionism and a resurgent antisemitism within the Soviet establishment.

The roots of these latter factors stretched back before the 1917 Revolution and how the thorny national question intersected with Russian radicalism. This issue was particularly vexing within the RSDLP as it was clear that workers' national identity was not being subsumed into a class identity. Jews, with their large organized socialist party, the Bund, made tactical alliances with the Mensheviks to favor an Austromarxist solution to nationality, a rather nebulous term that basically means that an individual has a right to declare their own nationality and would join in a federation of their respective national Marxist party. Lenin and the Bolsheviks responded to the Austromarxism's ascent with Stalin's famous pamphlet Marxism and the National Question. The resulting paper was the famous Marxism and the National Question attacked the principles of national personal autonomy and Austromarxism as a deviation away from Marx. The fundamental difference between the Bauer/Renner paradigm and Stalin's was that Stalin strongly asserted that a nation needs a territorial basis and must reside within a given territory and have a historical claim to that area. Austromarxism was thus a retrograde step according to Stalin as it diffused the strength of a Marxist political movement and would hamper the ultimate goal of Marxism after a revolution: the creation of a new socialist consciousness based upon class, not bourgeois concepts like nation.

Although the Bolshevik leadership tended to pay lip-service to Stalin's opus prior to 1917, Judaism occupied an ambivalent position in Stalin's formulations. The crux of the matter was whether or not Judaism counted as a religious practice or was an ethnic identity. If it was the former, then it was a vestige of an older order that the revolution would erase. But if Judaism was an ethnicity, then it would be protected by the post-Revolutionary order. Neither Lenin nor his compatriots ever really decided on definition over the other, preferring instead to craft tactical alliances with the Bund and like-minded Jewish radicals. Although Zionists made some tactical alliances with revolutionary forces in the charged situation of 1917, the Bundists found it much easier to ally with the Reds. A number of Jews also flocked towards the Bolsheviks for a number of reasons. The Red Army in many ways put a stop to the pogroms within Ukraine and the Bolsheviks eliminated the tsarist legal restrictions on Jews' movement and occupation. As the Soviet Union emerged from civil war, it found it needed a trained and educated bureaucratic civil service and many Jews, often Bundists, staffed these lower- and mid-level positions since Jews by and large had higher literacy rates than other members of the population. The Soviet state also encouraged Jewish cultural projects and held that all its citizens were equal before the law irrespective of ethnicity or religion.

It is tempting to view the 1920s through an idealized lens for Soviet-Jewish relations, however such idealizations are deceptive. Although the emerging Soviet state classified Jews as a nationality, they were never elevated to the status of "nation", a subtle, but important distinction as it implied a lack of permanent status for Jews as a people. Moreover, the Soviet state retained a degree of hostility towards the Zionist movement that was a holdover from the radical politics of the tsarist period. The special Jewish section of the Communist Party (evsektsii) was unlike other special ethnic sections in that it was explicitly temporary. By the late 1920s, the Stalin turn saw a partial victory of the perspective that Jewish was a religious practice and many of these special privileges (Terry Martin terms this Soviet-style Affirmative Action) granted to Jews were repealed. Stalin also arguably appealed to coded antisemitism in deposing of his rivals like Trotsky, who had a taint of a Jewish background.

Although Soviet Jews found the 1920s privileges curtailed under Stalin, the state did not openly hew towards an antisemitic line. Open statements of antisemitism were disciplinary offenses if reported to the state's authorities. The rise of the Third Reich also precluded a return to open antisemitism as German racism became lumped together as an ancillary component of Nazism's anticommunism within Soviet propaganda. Majority Jewish areas like Minsk openly embraced their identity as Soviet Jews as the ideological war between communism and fascism sharpened during the late 1930s. Despite this early form of Soviet patriotism, many Jews became victims of the Terror.

The Great Purges of the late 1930s had a large number of Jewish victims. Some of this was due to a covert antisemitism amongst security officials, but keep in mind many of the purgers were themselves Jewish in origin. Part of the reason for the number of Jewish victims is a function of how the Purges were conducted. If one had a questionable ideological background pre-1917, the odds of surviving the Purges lessened a great deal. Since many Jews involved in the Soviet state were former Bundists, they had a black mark on their biographical commitment to revolution that they could not erase like their gentile colleagues. Furthermore, informal antisemitism also meant that many non-Jews within the state saw Jews as outsiders, which was dangerous in the context of the circular firing squad of the Purges. Many of the mid-level managers when asked to name names by security forces would sacrifice outsiders before they would give up their inner circle.

The Second World War also saw an uptick in Russian antisemitism as the state no longer bothered to discipline antisemitic outbursts among the public. The life or death struggle with the Third Reich also precluded any acknowledgement by the USSR that antisemitism existed within its own borders because that would mean the Soviet Union shared common ground with its existential foe. A rekindling of antisemitism also received further sanction from the upper echelons of Soviet society as Stalin's paranoia directed itself towards Jewish elements within the USSR. Despite this postwar upswing, antisemitism within the USSR shared little with the antisemitism of Nazi Germany; Jews were still Soviet citizens and if they acculturated, they faced little formal discrimination by the state. That being said, Jews were an ethnicity that the postwar Soviet state possessed little love for, and the state was indifferent to complaints about antisemitism among the general Soviet public. Thus to be a Jew in the postwar USSR could be characterized as a begrudgingly accepted outsider.

Within this context, the Soviet's volte-face on the issue of a Jewish homeland makes more sense. Anti-zionism had been one of the consistent hallmarks of Soviet ideology since 1917, despite occasional tactical overtures. The brief sponsorship of the emerging Zionist state was an example of this type of tactical maneuvering as it did not lead to a reorientation of Soviet domestic policy towards the Judaism either in the USSR or in Eastern Europe. Despite Gromyko's championship of the a Jewish state in the UN General Assembly and the Czechoslovakian arms sale to Haganah, the Eastern bloc did not significantly ease up restrictions on Jewish emigration in this period and began prosecutions of Zionist groups for violating border controls. Soviet support for Israel was soon abandoned as it became apparent that the Arab world was a bigger prize to win. Soviet support for the Arabs filtered into preexisting ideological channels in Soviet foreign policy like its public support for anti-imperial struggles. Although the Soviet gains from supporting the Arabs often proved quite ephemeral, tilting towards the Arabs was something Soviet policymakers were more ideologically predisposed to accept than reversing decades of an often contradictory stance on Jewish affairs.

Sources

Abramson, Henry. A Prayer for the Government: Ukrainians and Jews in Revolutionary Times, 1917-1920. Cambridge, Mass: Distributed by Harvard University Press for the Harvard Ukrainian Research Institute and Center for Jewish Studies, Harvard University, 1999.

Bemporad, Elissa. Becoming Soviet Jews: the Bolshevik experiment in Minsk. Bloomington: Indiana University Press, 2013.

Gitelman, Zvi. A Century of Ambivalence: The Jews of Russia and the Soviet Union, 1881 to the Present. Bloomington: Indiana University Press, 2001.

Nathans, Benjamin. Beyond the Pale The Jewish Encounter with Late Imperial Russia. Berkeley: University of California Press, 2002.

Slocum, John W. "Who, and when, were the inorodtsy? The evolution of the category of “aliens” in imperial Russia." The Russian Review 57, no. 2 (1998): 173-190.

Wistrich, Robert S. Anti-Zionism and Antisemitism in the Contemporary World. New York: New York University Press, 1990.

I'd like to preface this by saying that the Soviet relationship with Jews was complex. My knowledge on the topic is primarily focused on Stalin's era, but I will contribute what I can.

Soviet foreign policy immediately after the Second World War advocated for the creation of a binational state in British Palestine first, or the partition of Palestine and the creation of an independent Jewish state if this proved impossible. In fact, the USSR was the first to legally recognize the sovereignty of Israel. By the end of the 1940s Stalin had toned down anti-Zionist rhetoric in the Soviet state media and even indirectly provided Israel with arms in the 1948 Arab-Israeli War. This period of cordial relations between the two states would last until the early 1950s, and was a result of Soviet pragmatism rather than ideology. Stalin had hoped that the new state could be swayed to the left politically and that it would signal the end of British influence in the region.

By the early 1950s it became clear that Israel sought to maintain close relations with the Western powers, thus Stalin's hopes that Israel could become a key ally in the region were unrealized. At the same time, official anti-Zionism in the USSR began to make a resurgence as Soviet politicians worried that close Israeli ties with the West would create internal tensions with the Jewish party leaders. Soviet ideological orthodoxy also labeled Zionism as a form of "racial imperialism," and with the pragmatic reasons for supporting Israel gone all that was left was ideologically impure support of a possibly unfriendly state. Around this time antisemitism in the USSR also became prominent despite the fact that it was officially banned following the Russian Revolution, and soon major controversies (many fabricated) such as the Doctor's Plot triggered a purge of Jews from the Soviet leadership and intelligentsia. At this point, supporting the Arab states was the more politically pragmatic option and internal resistance was silenced.

In essence, it was a combination of deep-rooted (though officially suppressed) antisemitism among the Soviet citizentry and leadership, ideological dissonance, and political pragmatism following Israel's overtures to the West that resulted in Soviet support shifting to the Arab states.

My answer is somewhat simplified for length, and there are many other factors that arguably influenced this shift in Soviet policy. These are the points that came up often in my readings in a history course I took on the Cold War last fall. Unfortunately I left the books in storage for the summer so I can't cite them. If there are any glaring errors or oversights please let me know and I will edit accordingly.