From one point of view the definition of "who is a Jew" doesn't rest with Jews. It rests with the anti-semites, and on who they choose to persecute or discriminate against as Jews. And they are typically more concerned about "bloodlines" than religion or even culture. Though I imagine that they'd be quite willing to extend the definition to include converts to Judaism and so forth, or people who exhibited cultural affinities, such as liking Klezmer music.

However the same kind of attitudes that show up as anti-semitism show up in regard to other minorities. Whether bigotry against Arabs qualifies as anti-semitism is a matter of semantics, but it's cut from the same cloth, and it appeals to the same shadowy place in the mind. I get the impression that for some people in parts of the right which have historically been anti-semitic, it has provided a convenient displacement mechanism, satisfying the same felt need - and it can be combined with expressions of support for Israel. A guilt-free alternative. Except it's not.