One of the easily overlooked aspects of OD&D is that all clerics are Lawful in alignment. At level 6 or below, Neutral clerics are possible, but those "of 7th level and greater are either 'Law' or 'Chaos'." Somewhat contradictorily, a later section of Men & Magic states that "there are Anti-Clerics ... who have similar powers to Clerics." As I read this passage, it means that "anti-clerics" are a distinct class of their own. They even have a different spell list from Lawful clerics (that includes no healing spells), as well as a unique spell -- finger of death. Likewise, they have no power over the undead as ordinary clerics do.
If one looks at OD&D from the perspective of AD&D or post-LBB editions of Dungeons & Dragons, anti-clerics are just another one of those oddities that doesn't seem to make any sense. But, as I've argued before, OD&D implicitly accepts a "fairytale Christianity," which it equates with Law and opposed by an equally storybook conception of Satanism/demon worship, which it equates with Chaos (often called "evil" in various parts of the LBBs). In such a context, anti-clerics are perfectly reasonable.
In my Dwimmermount campaign, there are anti-clerics, though I don't call them that, since, like the word "superhero" that OD&D also uses (and Chainmail before it), it somehow doesn't feel "right" to me. Actually, there's no single name for them, since these demon worshipers lack a unified organization, operating as part of dozens of different secret cults, each one dedicated to a different lord (or lords) of the Abyss. All, though, are wholly opposed to Law and devoted to the destruction of the present cosmic order, believing it to be a self-delusion and not at all representative of the great truth that only their masters know, namely that there are no gods and nothing awaits Man after death. Interestingly, elves believe the same thing, which is why many pity Man's brief pointless existence.
To date, I haven't done a great deal with anti-clerics in the campaign, mostly because the areas of the world the players have explored haven't really given me much opportunity to do so. The cult of Turms Termax remains the primary antagonist, although the Eld of Areon are shaping up to be important secondary opponents. This is fine by me. My feeling is that any campaign worth its salt should be larger than whatever the player characters are doing at any given time. The demon cults are out there, lurking in the shadows, and one day the party may cross swords with them. Or not -- that's for my players to decide.