Part of the problem is that there is not one "string theory" that makes a set of invariant predictions. Instead, there is a string theory "landscape" of conceivable string theory realizations, of which only one is correct in our universe.
There are some observations that would falsify all string theories if true. For example, if Lorentz invariance, or CPT symmetry were violated, or time ran slower than usual on Mondays. Similarly, if gravitons were discovered but turned out to be spin-3, odd parity particles, that would be contrary to the predictions of every string theory.
There is a conjecture (stated as fact by Lubos Motl among others), although I don't know just how rigorously it has been proven, that some form of SUGRA (the supergravity extension of supersymmetry) is the low energy effective theory of every viable string theory, and that some form of SUSY is the low energy effective theory of the non-gravitational component of every viable string theory. If this conjecture is true, then any proof that SUSY is not true would establish that string theory is not true (and likewise any proof that SUGRA's gravity extension is inconsistent with gravity would disprove sting theory).
Now, there is also a vast parameter space of possible SUSY and SUGRA theories, although a lot of it has been ruled out experimentally. It is certainly possible, in principle, to experimentally rule out the entire parameter space of SUSY and SUGRA theories, or at least a very broad class of them.
There are, for example, many experiments that show that SUSY particles must have masses not less than X. If one could determine generically that experimental evidence ruled out SUSY particles with masses more than Y, then as soon as X>Y then SUSY and SUGRA and string theory could be invalidated, so long as the conjecture is true.
For example, support that it is generically true that a SUSY theory with a lightest supersymmetric particle (LSP) mass of L has a neutrinoless beta decay rate of sqrt(L)*K. Then, if we knew that the neutrinoless beta decay rate was less than Z, then L<(Z/K)^2. But, suppose that other experiments had determined that L>(Z/K)^2. Then, SUSY would be ruled out, and a fortiori, so would SUGRA and string theory.
It might similarly be possible to determine theoretically, that no viable string theory in the landscape could have exactly the particle content of the Standard Model, because there is no appropriate Lie group that string theory can generate with those properties. It would then follow that there are either BSM particles with certain properties, or string theory is false. It might be much harder to rule out the BSM particles with those properties, but it would not in principle be impossible to do so.
