I suggest you check out my review of both this book and the Medoff book on the same subject but with a different conclusion (see the link below).
FDR and the Holocaust: A Breach of Faith - Print Edition
Lichtman and Breitman is essential reading on this topic, because it reminds us of the virulent anti-Semitism that pervaded the American community even through 1944 and later. It is difficult to believe how much of the worst anti-Semitism was voiced on the floors of both houses of Congress as late as 1944. The authors demonstrate in a rather convincing manner that had FDR appeared to be supporting policies that would send American boys to die in Europe for the purpose of rescuing Jews, he would not have been able to get Congress to approve re-armament or permit aid to England, all of which were essential to defeating Hitler. Even Jewish leaders were opposed to asking Congress to liberalize the immigration laws to permit more Jewish refugees into the country because they feared, with ample justification, that Congress would further restrict those laws, not liberalize them. Little tidbits like the opposition of the labor unions, which often had Jewish leadership, to permitting more refugees into the country are eye-opening. (The reason is that this opposition occurred in the early 1930's, when the unemployment rate in the US was close to 20%, and long before Chrystal Nacht, when we finally appreciated that the Nazi's were serious about their intention to murder Jews.)
So there is some merit to FDR's claim that openly helping to save Jews, or even the perception that he was doing so,risked undermining public support for the war effort against the Axis. On the other hand, FDR is known to have expressed some pretty nasty views about Jews on his own and perhaps did need that excuse for failing to act to save them.
The Jewish community was justified in supporting FDR politically, notwithstanding his refusal to help rescue Jews in Europe, because he was the only political leader likely to bring the US into the battle against Naziism militarily.
But on the ultimate question of whether FDR could have done more, and whether that failure was due to his inherent anti-Semitism, the Medoff book is more convincing. There was no reason why 75% or even 90% of the immigration quotas went unfulfilled, when there were hundreds of thousands of qualified applicants. No one can explain why permitting the use of existing quotas to permit refugees into the country would have created political problems for FDR. He had not created those quotas and all he would be doing in letting those quotas be filled would be to follow the law. Putting the blame on a bunch of blatant anti-Semites in the State Department (and there were plenty of those) does not excuse FDR, because they were his appointees, and he could have controlled the policies of his own cabinet. Moreover, Medoff (and even the Lichtman/Breitman book) reveal that on occasions FDR specifically approved or directed the restrictions on immigration levels or other rescue efforts. Too often, deliberate technical restrictions were created that were not justified and could only be explained by an intentional decision to prevent as many Jews as possible from getting to the US. For example, German Jews who were kicked out of their jobs and professions and were unable to bring any assets with them if they left Germany were denied visas to the US because of concern that, because they were unemployed and had no money, they would become public charges. If they still had family in Germany, they were denied entry for fear that they might be blackmailed into spying for Germany. The list of absurdities goes on and on.
There is also the question of whether FDR could have done anything anyway. Once WWII started, it was virtually impossible to rescue many Jews. But this does not apply to the situation prior to 1941, and perhaps many of the hundreds of thousands of Hungarian Jews who died after the Nazi's took control of that country in 1944 could have been saved. But we will never know, because no attempt was ever seriously considered.
There is also an incident that neither book covers--the offer of the Philippines to take 30,000 Jewish refugees and perhaps even more, during the years 1939-41. This is reported in a movie recently released, entitled "Rescue in the Philippines . . ." At that time, the Philippines was a US territory and needed US approval, which was denied. Finally, around the last week of November, 1941, the State Department approved letting in 10,000 refugees, but at the rate of only 1,000 a year. Two weeks later, the Japanese bombed Pearl Harbor, and not one Jewish refugee got to the Philippines under this arrangement. What could possibly have justified the State Department's refusal?
In the end, no one can prove that FDR used the excuse, that he could not let the public think that we were fighting a war to save Jews,to hide his own unwillingness to help, but the more one reads and thinks about it, the more one comes to the conclusion that FDR could have done a lot more but for his own inherent bias.
As stated above, read the Lichtman/Breitman book to get a better understanding of the frightening political and social climate of the 1930's and 1940's, but keep in mind that there is too much about FDR's actions (rather, inactions) that the book does not, and cannot, explain.