MWI certainly does make falsifiable claims. It's just that different interpretations of QM are extremely difficult to experimentally distinguish with any technology we are likely to have in the foreseeable future.
The only well-known interpretation of QM that is experimentally indistinguishable--in theory--from MWI is Bohm's Interpretation. And I think it would be a difficult row to hoe that Bohm's Interpretation is simpler than MWI, as Bohm's universe contains the MWI universe as a subset in the form of pilot waves that never collapse. I.e., the only difference between MWI and Bohm is that in Bohm, one of the "worlds" is granted reality, while the others are not.
Of course, there's nothing in MWI that grants all the worlds reality either. Someone can certainly remain agnostic about that. E.g., Stehphen Hawking has said that MWI is obviously true, though he is agnostic on the reality of the other worlds.
For more information on this, including a tutorial on all the linear algebra that you need to understand, see the book Quantum Mechanics and Experience by David Albert.
Another way in which MWI is falsifiable is that it would be proven wrong if those Chinese physicists were able to show a speed limit on the spooky action at a distance. MWI says that there is no such speed limit, since there is no spooky action at a distance.
Additionally, there are collapse theories that are much easier to test than consciousness-based ones. By "much easier", I still mean fantastically difficult, but should the day come that one of these collapse theories is proven correct, then MWI will clearly have been falsified.
You can set up complicated and impractical experiments (for us, but perhaps not for inconceivably advanced aliens) that can experimentally distinguish between MWI and other interpretations of QM. (With the exception of Bohm.) Do you really want to know all the gory details? They are rather obscure.
The executive summary is that it is in theory possible to determine whether a system is in a specific superposition of states, if you know exactly what superposition of states it might be in. So, let's say you wish to see if MWI is true vs a version of Copenhagen that relies on a conscious entity experiencing the results of a measurement in order to cause the wave collapse. If you have an accurate enough model down the very last particle of the entity's brain, and everything else the entity would have to interact with for it to perform an experiment, you can in theory then experiment on the entity to determine if it (plus everything it interacted with) is in the specific superposition of states that MWI would predict.
As I said, not very practical.
Well, David Deutsch argues that it might be practical when the day comes that we have AI's living in quantum computers. (And his experiment is also a bit different from what I described, and a bit more conceivable for us to do with technology we might someday have.) A problem there, is that naysayers will no doubt claim that the AI's aren't conscious.
The only well-known interpretation of QM that is experimentally indistinguishable--in theory--from MWI is Bohm's Interpretation. And I think it would be a difficult row to hoe that Bohm's Interpretation is simpler than MWI, as Bohm's universe contains the MWI universe as a subset in the form of pilot waves that never collapse. I.e., the only difference between MWI and Bohm is that in Bohm, one of the "worlds" is granted reality, while the others are not.
Of course, there's nothing in MWI that grants all the worlds reality either. Someone can certainly remain agnostic about that. E.g., Stehphen Hawking has said that MWI is obviously true, though he is agnostic on the reality of the other worlds.
For more information on this, including a tutorial on all the linear algebra that you need to understand, see the book Quantum Mechanics and Experience by David Albert.