To answer this, you need to define antisemitism. There are people out there who define antisemitism based on religion as something different, anti-Judaism. I think that's a bit of an artificial divide, but it is a common bit of terminology.

You also have to differentiate what's just xenophobia. Josephus wrote a book refuting a guy who spoke negatively of the Jews and Judaism. But that guy's views may've just been xenophobia, or a prejudice, or just ignorance. Mapping antisemitism onto it is challenging.

But that said, the earliest thing that fits well into later patterns is from the 4th century. St John Crysostom, a Christian theologian, wrote a series of homilies called Adversus Judaeos, or "Against the Jews". It contains early references to Jews being collectively responsible for Jesus' death, as well as other later tropes^1 . But there are earlier things you could fit into that, and later ones that might fit better.

1. There are people who think John Chrysostom was saying something less objectionable. But in my experience they're mostly Christians trying to avoid a prominent church father being a raging bigot, and if you actually read it those claims are difficult to support.