Whether (philosophical) naturalism is falsifiable depends on one's definition of the supernatural, and whether we can find a way to test, demonstrate, reproduce, etc. supernatural claims.
In theory, if we can demonstrate supernatural claims (and they'd still fall within our definition of supernatural), then that would falsify naturalism.
If we cannot demonstrate supernatural claims (perhaps because our definition of supernatural is very closely tied to not being demonstrable), then naturalism cannot be falsified, but also shouldn't be falsified, because it's rejecting claims that cannot be demonstrated.
But how would we demonstrate supernatural claims? The problem is that our observations are bound to the natural world. One could certainly conceive of a device that would allow us to clearly observe ghosts or other supernatural entities. One could also conceive of natural explanations for this, e.g. what we're observing may just be sufficiently advanced technology or natural phenomena we don't yet understand, but many people may nonetheless deem this to meet the burden of falsification, be sufficiently convinced of the supernatural and abandon philosophical naturalism. But the bigger question is: if we can clearly observe ghosts, would they still classify as supernatural? Opinions would probably be divided on this, and it's a crucial part of whether philosophical naturalism is and should be falsifiable.
Methodological naturalism, on the other hand, isn't a claim, it's a methodological foundation. As such, it wouldn't be subject to falsification. If supernatural claims are demonstrated, however, then we may reject methodological naturalism and extend our methodology to include those demonstrated supernatural claims.
Naturalism can mean one of 2 things:
Philosophical naturalism, which is a "worldview that holds that there is nothing but natural elements, principles, and relations".
Methodological naturalism, is a methodology/framework for acquiring knowledge, where you proceed as if no supernatural forces are interfering with the results, regardless of whether or not you believe in supernatural forces.
Both are based on the epistemology that claims should be e.g. testable, reproducible, verifiable and/or falsifiable, which supernatural claims have not shown to be (and arguably might not be by definition). Philosophical naturalism concludes that therefore we shouldn't believe them, whereas methodological naturalism concludes that we should exclude them from consideration when analysing natural phenomena.
A quote from armand's answer demonstrates why this makes sense:
if we ever managed to produce [measurable, reproducible evidence of ghosts], ghost would then have become a natural phenomenon, something we can reliably produce and control
That is to say: naturalists don't reject the existence of ghosts simply because they're supernatural, but rather because they don't meet our standards of evidence. Once they meet our standards of evidence, they would arguably (but perhaps not necessarily) be considered a natural phenomenon. Being a natural phenomenon isn't explicitly a criterion of our standards of evidence, but meeting our standards of evidence and being a natural phenomenon do go hand-in-hand.
All of scientific discovery (at least in mainstream science) is based on methodological naturalism. That is, conducting science as if no supernatural forces are interfering with the results, regardless of whether or not you believe in supernatural forces. When we see a domino being knocked over, we don't say "well, it must've been God" as the explanation, we look around for a natural explanation for what could've knocked it over.
If science weren't based on this, we'd still believe that supernatural forces cause lightning and make us sick, and therefore the fields of climatology and microbiology (and most other scientific fields) wouldn't exist.