The idea of a multiverse has a lot going for it. In quantum theory, the many worlds interpretation avoids some of the tricky problems of other interpretations. More generally, the question of why the universe seems to be tuned for life to exist is easier to understand if there are many universes - naturally we will find ourselves in one which is suited to life, but there may be many more which have no life. However, I have my doubts about the benefits of postulating a multiverse.
Karl Popper objected to Freudian psychology and Marxism because they had an answer to everything. In particular, if you criticised them, then it couldn't be because they were wrong - it was because there was a problem with your mind. You had been brainwashed by bourgeois society or were suppressing an event in your past. I see the many worlds idea as having a similar problem. Other universes exist, but things are set up in such a way that your mind can't detect them.
I would look on multiverse ideas more favourably if there was a way to travel between universes. I just don't buy the idea that things can become separated so that there can never again be any communication between them. Once it was thought that if mass fell into a black hole then it was gone forever. But then came Hawking radiation - the mass in a black hole is in fact gradually returned to the rest of the universe. I've also heard a new postulate of thermodynamics proposed saying that for any two systems there will be some possibility of interaction between them.
Then there's the question of a Deity. I'm not convinced that postulating a creator does anything to answer the question of why there is something rather than nothing, or why it seems to be tuned for life to exist. Some have put forward the multiverse as an alternative to a creator. I don't agree with this. In the Narnia books by C.S. Lewis there is the 'wood between the worlds', a place (in Aslan's country seemingly) from which each of the different worlds can be reached. If there is a multiverse then I would expect there to be a corresponding 'wood between the worlds'. Maybe, as in the Narnia books, travel between the worlds is more related to religion than to science. So I see the existence of a multiverse as supporting the possibility of a Deity, rather than arguing against it.