I'm not exactly an expert, but I'll still say none. In fact, the claim was even denounced by the American Historical Association, which in response to the book publicized that

The AHA deplores any misuse of history that distorts the historical record to demonize or demean a particular racial, ethnic, religious, or cultural group. The Association therefore condemns as false any statement alleging that Jews played a disproportionate role in the exploitation of slave labor or in the Atlantic slave trade.

First of all, despite the book's title, it's no secret that there were Jews connected to the slave trade. Not just in the Americas- also in any society in which Jews lived in which people owned slaves. This ownership was by and large representative of their number proportionately in the population. They definitely owned and traded slaves specifically in the Americas, though to different degrees at various times. This is something that has been known by historians basically forever (as far back as an early work of Jewish history from 1788!) and has never been kept a secret. (I remember seeing a fellow undergrad student at a conference who was excitedly telling her professor about how she had "discovered" that Jews were part of the slave trade, and he basically laughed and said, yeah, we know.)

That doesn't prevent The Secret Relationship from being essentially a calumny against the Jewish people, a book which Henry Louis Gates Jr of Harvard called "the bible of the new anti-Semitism" in his NYT review of the book. The claims that Jews were major or disproportionate parts of the slave trade, or that they owned far more slaves than the average white person, are often made by cherry picking data from intellectually valid sources (the book is very proud of its thousand-plus footnotes, though one review in The Atlantic notes how confusing and obfuscating the footnoting system is) and taking it out of context. Some data points which are used in the book are actually taken from books and articles which endeavor to prove (and succeed in proving) exactly the opposite- that Jews were a minute part of the American slave trade/ownership, at most proportionate to their (also then tiny) proportion of the population. In other cases, the book relies on information from biased secondary sources, as I'll get into below.

Any of the above links will contain good information to start with, as will this article by David Brion Davis, one of the historians responsible for the above AHA statement, and this article, which includes great quotes from both Davis and Eli Faber, who wrote a whole book responding to The Secret Relationship called Jews, Slaves, and the Slave Trade: Setting the Record Straight. However, since this is AH and this should really be a one-stop shop, I'll go into some of the reasons why the book is bogus and how it got the info it did:

• As mentioned above, a lot of facts and numbers were taken straight from articles which actually proved the exact opposite of the book's claim. These articles/books were used specifically because they were written by Jews, as the NOI believed that if the Jews themselves were writing it, then it would be more credible/believed (as in, "even you say it's true!"). Of course, "the Jews" by and large said nothing of the kind, or if they did it was as part of a very different argument.
• Some information was taken from works by Jews which did state that Jews were a large part of the slave trade or did their best to emphasize that fact. There were two kinds of such works: 1) Jewish academics who were trying to reckon with the darker side of Jewish history and went a bit overboard and 2) Jews from 150-200 years ago who were trying to emphasize the integral part which Jews played in American history and believed that showing Jewish participation in slavery would help. The second kind of issue is EXTREMELY common when reading most early American Jewish history and it is always necessary to try to read around it.
• As noted by reviewers, the book began with a thesis and cherry picked data to prove it, ignoring all relevant context. As noted by Davis in his article, far more slaves (by an order of magnitude) were owned by free black people in Haiti in 1789 than were owned by Jews in the American South in 1830, one of the years picked out by the book as being a time of major Jewish slaveholding. In general, the book listed numbers of Jews who owned slaves, the number of slaves they owned, etc while never even attempting to compare the numbers to those of slaves owned by white Christians, as they would have been dwarfed.
• In some cases, the book contains easily falsifiable lies- as pointed out in the review in The Atlantic, the claim that all of the distillers in Newport, Rhode Island (with distilleries being key parts of the North American slave trade) in the 18th century were Jews is untrue when one looks at the records.

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