tl;dr: It probably won't affect your career.
You would be in clear violation the academic boycott
Please read the guidelines document of the Palestinian academic boycott, which is the (most/single) relevant aspect of the BDS movement to your question.
In the document, you will note the position that:
Academic institutions are a key part of the ideological and institutional scaffolding of Israel’s regime of occupation, colonialism and apartheid against the Palestinian people.
Specifically, the following events, activities, or situations are in violation of the Palestinian academic boycott:
- International students enrolling in or international faculty teaching or conducting research at degree or non-degree programs at an Israeli institution.
I'm assuming you are not Palestinian, nor an Israeli whose enrollment in Israeli institutions is obviously exempted and not considered a violation.
It does not matter:
- What the contents of your research is
- What your political position is
- What the political position of your research supervisor/advisor is
You would be considered as benefiting at the Palestinians' expense and legitimizing their predicament
Millions of Palestinians are not able to travel to Palestine (especially the areas considered part of sovereign Israel), live there, study and work there, and particularly, undertake Ph.D. programs. You, however, will do so; and the symbolism is pretty powerful. This will be perceived by many as benefiting from the dispossession and ethnic cleansing of the Palestinians. Moreover, it is the Israeli state at whose will you may enter, and they - the Palestinians - may not; and your availing yourself of that possibility will be perceived as legitimizing this control, this power, Israel has.
Whoever becomes your advisor would be abetting your violation
An academic at an Israeli university who respects the boycott cannot be your Ph.D. supervisor/advisor.
"... but will this all affect my career?"
Only a minority of academics support the boycott - not because most academic oppose it (which they may or may not), but more because most academics have never given it much thought, and have at most heard about it in the media. I'm not basing this on statistics though, but rather on my impression and limited personal knowledge.
In addition, only a minority of those who support the boycott - again, in my belief and to my impression - would actively penalize, or distance themselves from, people who violate the boycott. And that is particularly true for a Ph.D. candidate, who can claim "I just went there, I don't accept Israel's policies nor the collaboration between the universities and the military etc."
However, if you are interested in academic activity in subfields or in institutions in which this minority is more common (e.g. in Arab countries; Middle-Eastern/Palestinian Studies; etc.) - then the chances are somewhat higher. Since I'm not in that situation (nor have I violated the boycott), I can't be more specific.
Even then, institutions in most countries will not condone discrimination against you based on your violation of the boycott, so the space for actions against you would be limited. In fact, in some places - it is people who endorse the boycott, or that are critical of Israel and US and European support for it, that are ostracized and penalized. The US almost passed a law making support for BDS illegal, and one of the states actually went ahead with it (although that state law was struck down as unconstitutional).
So your career will probably be unaffected. It will mostly be a matter of conscience you have to decide.
(Of course it will affect your career by your being immersed in the academic and social environment of an Israeli university, which will affect how you develop as a person and your perspective as a researcher. But that's not quite what you were asking.)