Which type of "anthropic principle" are you referring to? The "weak" version of the principle just says that naturally
if sentient observers exist, they must find themselves in a part of the universe (or multiverse) with suitable conditions for their existence to occur--conditions which may be rather "special" with regard to the larger universe/multiverse. It doesn't say that such conditions
must exist--we could very well imagine a universe whose laws were such that there were no sentient observers at all, the weak anthropic principle doesn't say it was somehow "inevitable" that the real universe would be a more hospitable one (the 'strong anthropic principle' does say this, and is thus harder to swallow if you don't believe the universe evolves according to natural laws which don't have any foresight or goal). But if we assume that some of the seeming "constants" of nature were decided in a random way by
spontaneous symmetry breaking shortly after the Big Bang, then even if very few possible values were compatible with the evolution of sentient observers, the many-worlds interpretation makes it natural there would be
some fraction of "worlds" where the constants allow for sentient observers, and the weak anthropic principle says that naturally we will find ourself living in this fraction, however small and non-representative this fraction may be. So in a way the MWI would make the "fine-tuning problem" go away (assuming the argument is correct that most values of the constants would be incompatible with intelligent life) by saying that even if it's a priori quite improbable the constants would take just the right values to allow for intelligent life, in a multiverse even the most improbable events are bound to happen in
some history, and as long as some such histories exist it should be no great surprise that we find ourselves in one of them.