There is a simple response to these concerns: Multiverses are not scientific theories in themselves. Rather, multiverses are predictions of certain theories of physics. The same theories that predict multiverses can make other predictions which are verifiable.
For example, take the theory that the universe is uniform on sufficiently large scales. There are galaxy filaments, huge thread-like structures made of galaxy clusters and superclusters. But if you zoom out, you will not find the galaxy filaments organizing into super-filaments, super-filaments organizing into ultra-filaments and so on. Instead, you will find that they are placed uniformly and randomly. That is a prediction we can verify.
But there is another prediction that can't be verified so easily: if the universe is uniform (and the curvature of space is non-positive), then the universe is infinite in extent.
While an infinite universe is more complicated in the sense that there is more stuff in the universe, it makes for a simpler theory with fewer parameters. If the universe is infinite, then I don't need to measure its size and shape, or measure our position relative to its center. In absence of evidence to the contrary, I will believe that the universe is infinite, because that is the simpler theory.
It is true that the scientific method requires hypothesizing, predicting, testing, interpreting, and theorizing, just like you see in middle school science fair experiments. But this is absolutely not true if you chop science into little pieces (say, into individual PhD theses). Science is not fractally structured. A single scientist can work entirely on theorizing without doing any experiments herself. Another scientist can work entirely on experimentation, and then just borrow interpretations from the other scientist. And if you take a single idea in science, it's unlikely that it would fulfill the roles of hypothesis, theory, prediction, and interpretation all at once.
In saying that multiverses are predictions, not theories, I'm repeating a point made by Max Tegmark in Scientific American. Tegmark also has a useful classification system for multiverses:
When talking about parallel universes, I find it useful to distinguish between four different levels: Level I (other such regions far away in space where the apparent laws of physics are the same, but where history played out differently because things started out differently), Level II (regions of space where even the apparent laws of physics are different), Level III (parallel worlds elsewhere in the so-called Hilbert space where quantum reality plays out), and Level IV (totally disconnected realities governed by different mathematical equations).Earlier I said that multiverses are predictions of certain theories of physics. Theories plural. Different theories predict different kind of multiverses. For example, the theory that the universe is uniform on sufficiently large scales predicts a Level I multiverse. The theory that there is no quantum wavefunction collapse predicts a Level III "many worlds" scenario. Certain theories of particle physics predict that the physical constants of nature will vary over very large scales, leading to a Level II multiverse.
Level IV is not predicted by any theory I know of. So maybe Level IV multiverses are unscientific. They can't all be winners!
Even though multiverses are predictions of certain theories of physics, the argument for multiverses that most people are familiar with is the anthropic principle. The argument goes that the reason our universe is so exceptional as to allow life is that there are many parallel universes and we only see the exceptional one. But while this is a satisfying result of multiverses, I don't think it is a very strong argument for multiverses. It doesn't even predict any particular kind of multiverse. This contrasts with the scientific arguments for multiverses, which are more specific in their predictions.