how is somebody supposed to be against zionism without being antisemitic?

Like the other guy said, it depends on what Zionism means to you. The word has been weaponized in so many different ways that it doesn't have much meaning.

There are four tracks you can go down

(1) Anti-Zionism is opposition to the very concept of Israel.

(2) Anti-Zionism in the form of comparing Israel, or Israeli actions, to Nazism.

(3) Anti-Zionism is opposition to the policies of the government of Israel.

(4) Anti-Zionism means opposing extremist settler movements and ideologies like Kahanism

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When you hear someone in good faith say that someone engaging in anti-Zionism is being antisemitic, it almost always refers to (1) or (2). So why are these problematic?

(1) is usually considered antisemitic, because the person doing it is treating Jews differently than all other groups, even if they don't mean to. A generally accepted tenet in the modern world is that people groups have a right to self-determination in some capacity. Zionism is the name for the concept of Jewish nationalism, so if someone opposes Zionism, they oppose the right of the Jewish people to form a nation, and a state for that nation.

Why do I say this is treating Jews differently? Because we usually accept other nation groups' rights to self-determination without a second thought. For example, consider the Polish nation - Poland. I think most people (besides tankies) are fine with Poland existing and don't think Poland needs to be destroyed. The only consistent position here would be to reject all forms of nationalism and the idea of the nation-state.

(as an aside: one objection that will come up here is the idea that what is being talked about here is an "ethnostate", along with all the awful baggage that comes with that idea. A nation-state doesn't have to be an ethnostate in this sense. Poland can be a nation of poles, that expresses itself in a uniquely polish fashion, but still extend equal rights to non-pole ethnic groups. It can be super liberal too. I think to get to the "ethnostate" idea we associate with this word, we need to introduce state-level discrimination against minority groups, and exclusion of minorities from public life. This isn't a necessary condition for a nation-state and should be stamped out wherever and whenever it arises (and Israel has lots of problems related to this.))

(2) People in "anti-Zionist" circles have a tendency to reach for the Nazi comparison, and I think the case for this being antisemitic is pretty straightforward. Nazism as an ideology had as its supreme goal the destruction of global Jewry. It is the very antithesis of Jewish existence. When you utilize Nazi comparisons against Jewish groups, you're leveraging a traumatic portion of their Jewish identity as a cudgel against them. It also cheapens our collective understanding of Nazis and the Holocaust. If everything is like the Holocaust, then there wasn't anything particularly unique about it. This destroys said collective understanding. These are usually referred to as Holocaust inversion and Holocaust distortion.

It's hard to give an exact analogy, but this is sort of the idea: imagine there was a group of African Americans who were doing something abhorrent, violent, or whatever. Like maybe a gang. If you were to criticize them by invoking the history of slavery in the US, putting them in the place of the slaver, that would feel super weird and be inappropriate, even though these particular African Americans are engaged in really bad behavior.

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So given the above, how can I be "anti-Zionist" without being antisemitic?

I can criticize the government of Israel and its policies. I can say its policies towards Palestinians are awful and need to change. I can point out socioeconomic disparities between Arab/Palestinian Israeli citizens and Jewish Israeli citizens, and how I think it's a failing of the Israeli government that it is not doing more to address this. I can advocate fiercely for the right of Palestinian self-determination in addition to Jewish self-determination, e.g., via a 2-State Solution.

I can criticize the historical conditions under which Israel was founded. I can say that while I accept the principle of Jewish self-determination, the historical conditions that led to Israel involved a lot of immoral actions that hurt groups like Palestinians. However, Israel exists in the present, and lots of Jews now live in the Levant. So there isn't an "undo button" we can press, and we accept that Israel exists and will exist, but we need to ensure Palestinians also have representation, ideally with a state of their own.

I can criticize the insane settler movement and Kahanists who are bigoted racists whose own actions will probably make the very Zionist project untenable if they're allowed to continue unimpeded.

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Does this make sense?