I think the other current answers are a little disingenuous. There were various riots and revolts of Arabs and Jews in the British mandate during the 1930's that caused strong backlashes against Jews in the region. A better phrasing of your question might be "How widespread was antisemitism in the Middle East before the establishment of the British Mandate," because antisemitism really takes off in that period.
Before that, Muslims and Jews got along relatively well. Under the Ottoman Empire, Jews were traditionally second-class subjects and not treated any better or worse than the Empire's Christians. After the Tanzimat reforms of the 19th century there were sincere efforts to elevate all of the empire's subjects to equal citizens. Nationalist aspirations of Ottoman Christians resulted in riots, massacres, wars, and ethnic cleansing and much animosity between the region's Christians and Muslims. Relations with the much smaller Jewish populations of the empire were serene by comparison.
The famous example of the first blood libel case in the Ottoman Empire, in 1840 in Damascus, was instituted by the French consulate in that city, in response to a missing French priest. This was under a brief period of Egyptian rule. The response of the Egyptian and Ottoman governments was to try and prevent this newly introduced idea from spreading.
After the Balfour Declaration and British occupation of Palestine Muslim sentiment quickly turned against Jews. Antisemitic tracts from Europe were rapidly translated and dispersed. This is why current Islamic Antisemitism looks EXACTLY like the virulent European Christian antisemitism of the 1920's, 1930's, and beyond. Muslims and Jews had previously gotten along fairly well, especially in comparison to how poorly European Christians treated Jews. Once Muslims began to turn against Jews because of the creation of the Mandate of Palestine, they quickly absorbed and directly translated the discourse of antisemitism into Arabic and other languages. Since that time, both Muslims and Jews have actively tried to erase the memory of the Ottoman era.
A couple of good books on the topic are "Ottoman Brothers," by Michelle Campos on the Christians, Jews, and Muslims of early 20th century Jerusalem, and Mark Mazower's "Salonica, City of Ghosts." Salonica, in modern day Greece, was long an incredibly vibrant, mostly Jewish city under Ottoman rule. Both can get you pointed in the right direction.
The hardcore antisemitism you see for instance in the Charter of the Hamas (f.e. blaming the Jews for the French Revolution) was picked up from «Radio Zeesen». It was a German project starting in 1938 to broadcast with stations in multiple dialects of Arabic, and later also in Persian, in Urdu and in Turkish until the very end of the war. It was modeled on the Italian «Radio Bari» available from 1932: Aside from the news they had programmes with comedians, with musicians, with the clergy; The transmitters were given out for free to cafés, hookah bars and to other public venues which could be reached from the German embassies. After the war the Muslim Brotherhood would pick up their arguments and cite the ahadith they dug up first.
source: «Islamischer Antisemitismus und Deutsche Politik» by Matthias Künzel
Of course it's not that life for the minorities was significantly better than in Europe before. The pogroms, the calumnies and the attempts at forced conversion were just the same as in the West.