Antisemitism is as old as Judaism is, I'm sure. I think the answer lies in how very different Judaism is from other religions, originally and all the way to present times, and also because of people's lack of knowledge of it. I should also say that many people are not antisemitic, so it might be more reasonable to ask why some people hate Jews.
First, Judaism was the first monotheistic religion, as far as we know. Prior to Judaism, people worshiped many gods, so this was a paradigmatic shift. An alien concept disrupts people, and usually not in a good way. Furthermore, most religions had power structures, for example, various priests to promote various gods. When Judaism came into being, the state was usually part of that power structure. So, for those in charge, rulers and a hierarchy of priests, this was a threat to their power and to their wealth. For the common people, this was upsetting because their gods were what they had been relying upon and sacrificing to, for thousands of years. None of this was likely to win friends and influence people! It is not particularly surprising that Jews were hated.
Second, once Judaism began, there is ample evidence in the Torah that Jews promptly began to separate themselves from those around them, in their practices and rituals, focusing on being as different as they could possibly be. If you are going to start a new religion, then it makes sense that you would want to make up new rules for the religion and insist upon your people following those rules. Not eating pork, not eating shellfish, not working one day a week, keeping meat and milk separate - all of these would have been practices that were emphatically different from the culture around the Jews. Did the people around them feel that they were now being regarded in contempt? This is entirely possible. The Jews argued right from the beginning, at least as far as written history is concerned, that they had been "chosen" by God. Imagine what that would make you feel like if you were not Jewish. Remarkably, these differences have persisted throughout all of Jewish history, to the present day, and we all understand that people really do not like differences very much. Nor do they appreciate it if they have any sense at all that these differences are meant to make someone feel superior to them. In fact, that is not the purpose of all these rules or even the purpose of being "chosen," but sometimes I think this is how it comes across, and this can certainly nurture dislike or even hatred.
Judaism existed in this state of "differentness" for a few thousand years, and then Christianity burst upon the scene. Christianity, too, as a new religion, insisted on being different, although Jesus Christ clearly maintained many Jewish practices, the Last Supper being evidence of his continuing to observe Passover, for example. But antisemitism was exacerbated by the New Testament, which holds Jews largely responsible for the death of Christ. For one group to be responsible for the messiah of another group is not going to foster a harmonious relationship between the two groups. And for one group to deny the messianic status of another group's leader is not likely to do so, either. I do not know at what point in history people began to spread rumors about various Jewish practices, for example, that children were sacrificed at Passover, but rumors like this persist to this day, and of course, if people have no knowledge of Judaism, they might very well believe these things and consequently fear and hate Jews.
It is astounding how many people know little or nothing about Judaism. But since Jewish people never proselytize and we are such a tiny group, perhaps this is understandable. Judaism has no "Outreach" the way Christian denominations do. Conversion is actually discouraged, probably a result of the unpopularity of Jews throughout history, making conversion attempts problematic and even dangerous. And Jews represent 0.2% of the world's population, which means many people have little or no exposure to Jewish people and practices. I have had many students ask me many questions about my religion because I am the first Jewish person they have ever spoken to, and I live in a major metropolitan area of perhaps a million people. What we do not know, we fear and even hate, and if we have no opportunities to know, there is no means of learning.
As the world becomes a smaller place, I would like to think that people will get to know more about other religions, not just Judaism, and gain some insight and respect for the idea that there are many paths to righteousness, that no one has the one true way. As people are dying all around us in the name of religion, we all simply must do better.