Short answer: we can (occasionally) check some elements of the Anekdota. For instance, a miaphysite (also traditionally called “monophysite”) hagiographic work by John of Ephesus confirms that Theodora indeed had been a performer-prostitute before marrying Justinian (which is, after all, one of the most famous allegations of the work). However, the general rule is that we don't have anything to crosscheck Procopius allegations, which means, in turn, that almost every possible position has been taken on the fiability of his work; traditionally, as laid out by Averil Cameron in her Procopius and the Sixth Century, historians have tended to accept the Secret History as a globally reliable source, even when it was considered to be distasteful.

Three recent interpretations include that of Anthony Kaldellis in Procopius of Caesarea: Tyranny, History, and Philosophy at the End of Antiquity (Procopius as a subtle dissenter, who basically told the truth about a tyrannical regime—it is a forceful and, it should be said, idiosyncratic restatement of the traditional vue), that of Av. Cameron (Procopius as a bigoted, misogynistic conservative who libelled the imperial couple) and even the original and unpublished argument of Henning Börm (Procopius knew that a conspiracy could overthrow the imperial power, and therefore prepared an exaggerated document in order to clear himself of all accusations of proximity with the future-former imperial power). To give you an idea of the reception of these hypotheses to deal with Procopius, that of Börm is, well, probably not very well-known (and since it is smart but ultimately unprovable, it probably never will, except in footnotes), that of Cameron is relatively mainstream, though it has not eliminated the previous one (i.e. reluctant acceptance because, eh, that's something to work with).