A few points: first, the version of the multiverse that tries to avoid the obvious conclusions from the fine-tuning of physical constants should not be confused with the many-worlds interpretation of quantum mechanics, which is trying to make quantum mechanics look more like classical mechanics by avoiding the collapse of the wave function when an observation is made. (I find interpretations of quantum mechanics pointless, because having learned my physics in the reverse of the usual order -- quantum mechanics and general relativity first, since they impinged on my mathematical research, then classical physics from being asked to teach Calc III and Diff Eq -- I see no point in trying to make quantum mechanics look like classical mechanics. We understand why large ensembles of quantum mechanical particles act classically because we understand quantum decoherence and have formulated statistical mechanics -- though it's not quite fair to say we understand statistical mechanics.)
Second, and this should be pointed out to any atheist invoking it: Occam's razor favors a single transcendent deity as the source of our ordered universe over the existence of a vast, perhaps infinite, number of universes most of which have physics that doesn't lead to the interesting phenomena (like us) that we observe in the one universe we know by observation exists. Moreover this multiplicity of other entities are just as unverfiable by observation as is God. If they could be observed, they would, perforce be part of our universe, not part of another. If the atheist really cannot stand the idea of a personal absolute as the source of being, he or she would be better off becoming a Taoist, and at least having a single, self-less absolute as the ground of being, rather than adopting a non-empiricially verfiable position that falls to Occam's razor when compared with monotheism.