I should start by mentioning that Thucydides, who belonged roughly a generation behind Herodotus , wrote history in a much more analytic and rigorous manner than his predecessor did. Indeed, he seems to criticize Herodotus for being too fanciful in his claims, when he writes about "the prose chroniclers, who...
I should start by mentioning that Thucydides, who belonged roughly a generation behind Herodotus, wrote history in a much more analytic and rigorous manner than his predecessor did. Indeed, he seems to criticize Herodotus for being too fanciful in his claims, when he writes about "the prose chroniclers, who are less interested in telling the truth than in catching the attention of their public, whose authorities cannot be checked, and whose subject-matter . . . is mostly lost in the unreliable streams of mythology" (Book 1, page 21. See page 47 in Penguin Edition). I don't write this to criticize Herodotus, but I would respectfully suggest that we should be careful not to overlook Herodotus's flaws as a historian. Classical historians did have standards (even if those standards were largely set by Herodotus's successors), and by those standards Herodotus did fall under criticism, and this criticism did not only come from modern historians.
The major theme of this question lies in the interplay of religion and critical analysis, and how to evaluate the two against one another. Indeed, I would note, beyond the strict appearance and role of the gods in human affairs, his entire narrative is rife with mythological overtones. His treatment of Cyrus the Great in particular reads as a kind of hero's myth, suspended somewhere between fact and fiction (and it's difficult to tell where the one ends and the other begins), not to mention the earlier exchange between Croesus and Solon, which reads as a kind of moral story about the superiority of humility over hubris. In addition we can give mention of Book Two, where he gives a lengthy digression concerning Egypt that often reads as pure exoticism in its tone, and this digression carries on across pages, as it moves from the people and the culture to the many strange and fantastic animals that live there (including, rather memorably, the phoenix).
Of course, all that being stated, I have to agree in that it's problematic to judge Herodotus by standards that would not have existed yet, but in this case the issue isn't so much that he was a Greek polytheist who believed in the gods, but rather, I'd argue that the deeper issue was the simple fact that the field of history, as we know it today, and indeed even as it would later come to exist in the Greek and Roman world, did not exist yet in the time that Herodotus was writing. Indeed, part of the difficulty in analyzing and evaluating Herodotus as a historian was that he was in his own way as much a storyteller as he was a historian. He opens the Histories with the following statement: "Herodotus of Halicarnassus here displays his inquiry, so that human achievements may not become forgotten in time, and great and marvelous deeds . . . may not be without their glory; and especially to show why the two peoples fought with one another" (Penguin Histories, Book 1, page 3). As far as mission statements go, we can see in his very intentions and sensibilities elements of the heroic and even the epic at play.
That's one of the tensions running across the Histories—the inner conflict between Herodotus the Historian and Herodotus the Storyteller, and how, as a modern reader, you set about negotiating between the two. He darts back and forth between lengthy digressions about far off and (from a Greek mindset) exotic places and larger-than-life heroic individuals, and then you have the much more concrete narration of events which would line up more with what we'd expect from conventional histories (and this is most clearly expressed in his treatment of the Persian Wars themselves). His treatment of the divine adds a further complication to this but could be understood along much the same lines (though perhaps not without forgetting, as the other writers before me have stressed, the cultural and religious context in which Herodotus lived). In the end, I think it must be said that one should be careful not to read Herodotus through modern eyes.
Ultimately, I'd say, for me personally, the core conclusion is this: Herodotus, as the first real historian, represents the foundation on which the field of history rests. This makes him very important, but it also means that the genre will have evolved past him, as later generations build upon his work. Thucydides is particularly important in this, as the first to apply more modern standards of academic and analytic rigor, but as I said before, it's problematic to fault Herodotus for not matching later standards, considering those standards did not exist yet.
Citation note: this essay made reference to the following editions:
Herodotus, The Histories (Penguin Classics). Translated by Aubrey de Sèlincourt with revisions by John Marincola. London: Penguin Books, 2003).
Thucydides, History of the Peloponnesian War (Penguin Classics). Translated by Rex Warner. New York: Penguin Books, 1972.