Actually, anti-semitism not only refers to Jewish people, but also to Arabic people as they are both semitic ethnicities.
Discrimination and prejudice against Jewish people is specifically referred to as anti-semitism. Anti-semitism can be both racialized (against ethnic Jews), and religious (against followers of Judaism). Is anti-semitism a problem today? Is it widespread or not something to worry about anymore? Your answer should reflect how you perceive the climate of anti-semitism globally.
This is actually not quite true. While Arab people are semites (which refers not to ethnicity, but to language), the term "anti-semitism" was popularized specifically to reference hatred against Jews. This was done primarily by Wilhelm Marr in the late 1800s as an attempt to find a more scientific-sounding word for Judenhass. He wanted something that carried not just a religious connotation, but a racial connotation, as well.
Until another word for discrimination against Jewish people becomes more popularized, anti-semitism is okay to use when discussing hatred or prejudice against Jews, as that it's how it's been used for over 100 years now.
And that's a fair point, but it's a microcosm of the broader problem that the only descendants of Shem who are linguistically included are those who are Jewish.
Anyone who does anti-racism work knows how quickly the accusation that a person is just a puppet of a Jew shadow cabal will come out. The anti-Semitic fear is a deep cultural one that has a lot of antecedents and causes. Even well-meaning people often wonder why Jews have seemed to be so successful in certain fields like entertainment and finance. Unfortunately, many of us do not have the kind of training to recognize how everything from small sample sizes to geography to specifics of the journey to whiteness all play into that impression of Jewish success. And it is very easy to turn from admiring a group's success to treating that group as a monolith to then demonizing and suspecting them.
However, anti-Semitism is a tough topic to discuss because it has all too often been launched at anyone who criticizes Israel. There are certainly people who oppose Israel for anti-Semitic reasons. But the region is full of ethno-religious and ethno-linguistic resentment. Israel has itself demonstrated anti-Arab, anti-Muslim and anti-African sentiments, even internally (look at the treatment of Sephardic and Ethiopean Jews in Israel). So it's hard to measure anti-Semitism in a realistic way because so many anti-Semitic activists have distorted that data by including criticisms of Israeli foreign or domestic policy in it. And you can't entirely blame them either, because there does seem to be a unique vitriol around debates about Israel and a unique paranoia. The fact that the region sees Jews facing Arabs, Turks and Persians makes the whole thing alien to many Europeans and Americans and thus promotes a lack of empathy for or understanding of all the players in the region and their actual needs.