How does Anti-Semitism relate to the modern State of Israel?

Answer 1

Anti-Semitism causing Anti-Zionism
This should be self-evident. The Logic goes thus: Anti-Semites oppose any Jewish aspiration to freedom and/or power. Zionism, the Jewish Nationalist philosophy on which Zionism was based, promotes Self-Determination for the Jewish people which is an aspiration to freedom and power. Therefore, Anti-Semites oppose Zionism. Additionally, many people who are Anti-Semitic see Jews as running some sort of international cabal to strip power from everyone else. Equipped with a country, who knows what further havoc Jews could cause.

Anti-Zionism causing Anti-Semitism
Since Israel is the "Jewish State" many of its political policies are seen by those with little knowledge of Judaism to be part of Judaism. Additionally, they note that Jews outside of Israel do not criticize these actions and often openly support them. This leads to political Anti-Zionist sentiment becoming manifest in religious and ethnic Anti-Semitism. Many of the leaders of Muslim majority states have either said Anti-Semitic statements openly or have funded the circulation of Anti-Semitic literature (such as the Protocols of the Elders of Zion) as part of their demonization of Israel and its citizenry.

Note: While there are certainly those who say that Anti-Zionism is a form of Anti-Semitism, this particular answer uses the term Anti-Semitism to refer to exclusively anti-Jewish beliefs like: Jews are Christ-killers, Jews are descended of swine and monkeys, Jews control the economy, and other such nonsense which existed prior to and has nothing to do with the State of Israel.

Answer 2

By What They Believe In. And How Strong They Feel About It.