In his book on ancient Rome and Jerusalem, Martin Goodman suggests that it was the Romans who first became 'anti-semitic', demonising the Jews in order to justify their their overly harsh response to Jewish rebelliousness in 66CE. Their campaign to put down the rebellion ended with the complete destruction of the Jewish city four years later, and a subsequent diaspora that hasn't been matched, in terms of impact, by any other.
Jewish restiveness was apparently not the result of a general hostility to Roman overlordship, it was more 'reaction to maladministration by a low-grade governor'. The destruction of Jerusalem by Titus seems to have been completely unplanned, a spur of the moment decision by a general fed up with the toll taken by a stubborn, do-or-die phalanx of Jewish fighters.
Yet there's likely to have been another important factor in the decision to raze Jerusalem. Titus was the son of Vespasian, who, after the tumultuous year 69CE in Rome [which saw three caesars come and go], was in the running to become emperor himself. A glorious defeat, even an annihilation of the Jews [who had humiliated Rome by destroying the equivalent of a whole legion, an even without precedent in Jewish history], would clearly improve Vespasian's chances of gaining the laurel crown.
Before this rebellion, Jews were tolerated and even respected at Rome. Herod the Great managed to gain the patronage first of Mark Antony, then of Octavian, and his great building projects in Jerusalem matched the scale of those of Augustus in Rome. Later, Herod's grandson, Agrippa, gained the affections of Tiberius, as well as two later caesars, Gaius [Caligula] and Claudius. Due to their patronage, he was able to rule over even more territory than his grandfather. All this changed after 70CE, and the new Christian sect, keen to distance itself from the Jews and to gain the favour of the all-powerful Romans, found it highly politic to ratchet up their own anti-Jewish attitudes, blaming the Jews for not sufficiently supporting their Man in Heaven, and even for conniving at his death.
Could anti-semitism, that monumental Western negative force, really have sprung from such arbitrary beginnings? Well, yes and no. More no than yes, I'd say. Or rather, its beginnings were believable enough, but its burgeoning growth would have to be sustained by other factors.
One factor was the burgeoning growth of Christianity itself. At the time of Christianity's acceptance as the official religion of the empire, Augustine, that charming intellectual and sensitive soul, a truly worthy Church Father, argued that the Jews shouldn't be converted to the True Faith, but kept in their parlous state of subjection as an example to all of the wrong path taken. Such sentiments testify to the fall of Jewish stocks in society under the Romans in the centuries following the fall of Jerusalem.
The Jews themselves were much more tolerant of Christians than vice versa. They were accustomed to break-away versions of their faith, and the gospels, if any of them are to be trusted, provide proof of the more or less good grace with which Jesus' apostate preachings were tolerated.
This is a sketchy account of the origins of anti-semitism, but its growth as a force in the west would require much further analysis, some of which is given in Niall Ferguson's The War of the World, though his book really only covers the matter in detail from the nineteenth century. Why were all Jews officially expelled from England in 1290, from France in 1322, from Spain in 1492, and so on and on? Clearly they were the victims of their own success to some extent. Ferguson gives the figures for the nineteenth and early twentieth centuries: the Jews were vastly over-represented not only in business and banking, for which they have been traditionally reviled, but in the arts and the sciences, in academia and politics, in every area of endeavour that demands a bit of brainpower. It's likely that this over-representation reaches centuries back. Expelling the Jews [and confiscating all their worldlies] was always a good ploy for cash-strapped monarchies needing to shore up the support of their subjects, but this only partially explains the phenomenon of anti-semitism, which is wrapped up in notions of ethnicity, in-groups and out-groups and, later, toxic concepts about race.
I might go into this in more detail in another post, but then again I might not.