At its core, Zionism is the pursuit of Jewish self-determination.
Zionism and the establishment of an Israeli state is the global Jewish community exercising the right to self-determination. To deny Jews of this right to control their political future is inherently anti-Semitic. Nobody would argue that the UK, France, Pakistan or Thailand do not have the right to exist as states. To single out Israel is a manifestation of anti-Semitism. Therefore, anti-Zionism is tantamount to anti-Semitism. The UN General Assembly voted to recognise the state of Israel 72 years ago. The fact that we are still debating its right to exist after all this time is simply a manifestation of anti-Semitism.
[Counter 1] Anti-Zionism does not seek to deny Jews the right to self-determination. It merely disputes the right of the state of Israel to exercise that self-determination at the expense of the Palestinian community. Anti-Zionism, in the purest definition of the term, is the objection to Israeli expansionism and the suppression of the Palestinian people. There is nothing anti-Semitic about it. Nor does it deny Israel's right to exist. It is simply the promotion of equality and Palestinian rights. [Counter 2] Even if anti-Zionists did question Israel's right to existence, it wouldn't be anti-Semitic. Not all ethnicities and religions have the automatic right to a nation-state. The Kurds, Scots, Catalans, Kashmiris, Tibetans, Lombards and Uyghurs do not have a state. They have a right to exist in a pluralistic and tolerant society. But they do not have the right to demand the formation of a nation state. 
[P1] Zionism is the name given to the movement calling for Jewish self-determination. [P2] To deny Jews the right to self-determination is anti-Semitic. [P3] Therefore, anti-Zionist is simply a manifestation of anti-Semitism.
[Rejecting P1] Anti-Zionism does not criticise the Jewish right to self-determination. [Rejecting P2] Even if it did, it wouldn't be anti-Semitic.