First let us recall why string theory is attractive. As of now, we have two experimentally verified, but mutually incompatible theories describing fundamental physical phenomena. The standard model of particle physics is a quantum field theory describing all the elementary particle interactions except for gravitation. General relativity is a classical theory describing gravitation and classical electrodynamics, but none of the other fundamental interactions in the realm of the standard model. The fact that it is a classical theory also tells us that it is necessarily an approximation of reality.
String theory is a quantum theory which in certain limits can be described by quantum field theories similar to the standard model, and in other limits by classical gravitational theories akin to general relativity. It seems therefore that string theory offers a realm to unify our two apparently incompatible descriptions of fundamental physical phenomena.
Now there are two issues concerning the falsifiability of string theory.
The first one is common to any theory of quantum gravity. Very generic arguments indicate that "quantum gravity effects", i.e. effects that would mix quantum physics and gravitational physics, for which neither the standard model nor general relativity would be appropriate descriptions, occur at energies well beyond experimental reach. Because of this, there is no serious hope to test experimentally a theory of quantum gravity which is consistent with the standard model and general relativty. Of course these arguments are not water-tight, and we can imagine various scenarios where quantum gravity effects would kick in just at the limit of our current experimental reach, but such scenarios are necessarily unlikely. Let us emphasize that this has nothing to do with string theory, and is a problem common to any theory of quantum gravity that does not obviously contradicts what we already know about Nature. Also, we have computational methods allowing us to extract the physics from solutions of string theory. So should we have experimental data of physics at high enough energies, we would be able to rule out of confirm most of the solutions of string theory. Therefore this first issue is really about our limited experimental reach.
The second issue is specific to string theory. While we do not understand the theory completely yet, there are good indication that the space of solutions of the theory is very large (the so-called "landscape"). For instance, in the limit of low energy and zero gravitational coupling, where in principle we would expect to recover the standard model, one can find a huge variety of field theories, most of which have nothing to do with the standard model. There is therefore a feeling that "anything goes" and that there is no way of explaining from string theory the low energy physics we are used to. When you look at the details, it is in fact not true that anything goes, but there are still a large variety of field theories that can be obtained in this way, and it is not clear how the standard model would have been selected over all the other possibilities.
There is nothing we can do about the first issue. The best we can do about the second issue is to further our understanding of string theory and understand better its space of solution.
It should be also emphasized that confrontation with experiment, while of crucial importance, is not the only way we have to construct physically relevant theories. Einstein's theory of general relativity was confronted to experiment only after it had been fully formulated. Einstein devised it using consistency requirements, essentially requiring the compatibility of classical electrodynamic and gravitational physics. The development of string theory has been very much in this spirit, and string theory has passed an amazing number of very non-trivial consistency tests (anomaly cancellation, consistency of the web of dualities, etc...). The fact that consistency constraints seem to fix a unique form of the theory is an encouraging sign that it should have something to do with reality.