So, this is a rather difficult matter. Did Harry Truman say things that, by our standards today, we would consider anti-Semitic? Yes. His diary reveals as such. In a diary entry concerning United Jewish Appeal and its requests to put pressure on Britain to allow the entry of Jewish refugees into Palestine, he referred to Jews as "very, very selfish". He also claimed Jews have "no sense of proportion nor do they have any judgment [sic] on world affairs." Now, these remarks may be rather confusing given Truman's record including recognizing the State of Israel against the advise of the State Department, as well as having Jews as business partners, political allies, and close personal friends. As a senator, before being selected as Franklin Roosevelt's next vice president in 1944, Truman delivered speeches bringing attention to the plight of European Jewry.
Today — not tomorrow — we must do all that is humanly possible to provide a haven and place of safety for all those who can be grasped from the hands of the Nazi butchers. Free lands must be opened to them. Their present oppressors must know that they will be held directly accountable for their bloody deeds. To do all of this, we must draw deeply on our tradition of aid to the oppressed, and to our great national generosity. This is not a Jewish problem. It is an American problem—and we must and we will face it squarely and honorably.
It is important not to refer to anti-Semitism and anti-Semites as something monolithic. They come in all forms, with their feelings and words coming from any number of places. The anti-Semitism that Truman exhibited in his diary cannot in any way be likened to the anti-Semitism exhibited by contemporaries like Henry Ford or Father Charles Coughlin. Anti-Semitism would often come from a place of xenophobia, where Jews embodied the foreign immigrant that would take the jobs of native-born Americans. Then there is more chimerical anti-Semitism, which involves the hatred of Jews for any racial or religious characteristics. It's important to keep in mind that Truman was a small town boy from Missouri, and that making generalizations and ascribing certain social characteristics to large groups were not uncommon throughout the United States. It can even be argued that this specific expression of anti-Semitism in Truman's diary can be attributed to a personal hostility. When United Jewish Appeal made the aforementioned request, it was Henry Morgenthau who reached out to Truman. Morgenthau was one of the country's most prominent Jewish figures who had served as Franklin Roosevelt's Treasury Secretary. It was well known that Truman had a strong personal dislike for Morgenthau at this time, as they had a bit of a falling out after 1945. Therefore, it is altogether possible that the anti-Semitic rant is less of a representation of Truman's beliefs and more of a snapshot of his relationship with Morgenthau at the time.
For additional reading on American anti-Semitism, you can look at Till Van Rahden's article “Beyond Ambivalence: Variations of Catholic Anti-Semitism in Turn-of-the-Century Baltimore” published in American Jewish History, John Higham's article “Anti-Semitism in the Gilded Age: A Reinterpretation", as well as Richard Hofstadter's book The Age of Reform: From Bryan to F.D.R.