In order to falsify archetype theory, you would need something measurable
that is a consequence of the archetype theory.
So you would have archetype theory implies phenomena A, B and C.
Now if you would find that A, B or C is not the case, then archetype
theory is not true.
The problem is how to make this stick to archetype theory as a whole.
According to the wiki-page: "Being unconscious, the existence of archetypes
can only be deduced indirectly by examining behavior, images, art, myths,
etc. They are inherited potentials which are actualized when they enter
consciousness as images or manifest in behavior on interaction with the
outside world."
So if archetype theory only says that there are archetypes, then in order
to prove the theory as a whole false, you'd have to prove every proposed
specific archetype wrong.
I don't see a way to prove the theory as a whole wrong in one go. But I'm
not an expert on archetype theory.
There are even problems for proving a specific archetype wrong, because
they can only be deduced indirectly. Therefore you would need to formulate
specific implications of the existence each specific archetype, and check
for those. Take the God-archetype for example. What would the implications
be of the existence of that archetype? Is it sufficient to find that all
known cultures have gods? Does it preclude the existence of atheism? Does
it mean that everybody is susceptible the believing in the existence of a
god?
I suspect that neither archetype theory as a whole, nor the specific
archetypes are formulated rigorously enough to call them scientific in the
Popperian sense.