I've spent a lot of time thinking about this subject (in fact, I studied string theory at Columbia which is where Peter Woit is). The main issue that people have with string theory is the lack of testability / falsifiability. I think there's definitely some issues there, but I think blaming string theory is not quite right.
There are two reasons why string theory is tough to test. One of them, and the most obvious one, is that the energy scale where the strings exist is super high (likely at or near the Planck scale). This means that if we wanted to detect the direct signatures of strings, we'd have to create an experiment that is sensitive to these energy scales. This seems to be completely infeasible (people talk about particle colliders having to be the size of our solar system etc.) Now, interestingly this is not actually only a problem with string theory but rather with any theory of quantum gravity. This is because the fundamental scale of a theory is usually determined by combining its fundamental constants in some way as to produce an energy scale, and any theory of quantum gravity must somehow contain Planck's constant, Newton's constant of gravitation, and the speed of light. Taking these together gives you the Planck scale.
So, in other words if the reason you don't want to study string theory is that it only is testable at the Planck scale, then what you're really saying is that you don't think we should study any theory of quantum gravity. This is, I think, way to extreme of a position.
Now, interestingly, string theory is actually more than a theory of gravity, so unlike something like e.g. loop quantum gravity which is only a theory of gravity, it's conceivable that string theory somehow within it contains information that "trickles down" to lower energies and thus could potentially be testable at something like the LHC. This leads me to the second reason why string theory is very difficult to test.
It turns out that the equations of string theory are more-or-less unique at high energies but that as you start lowering the energy at which you probe the theory, multiple distinct solutions emerge. These solutions turn out to have a very nice physical interpretation: they are the different ways in which we can compactify the extra dimensions of string theory. Regardless of this physical interpretation, the fact remains that there are many many distinct solutions of string theory at low energies, and in order to make predictions that are falsifiable, we need to know which of these solutions we're living in. This is where the crux of the problem lies. It turns out that there are so many solutions of string theory that we cannot even in principle go through them one at a time to see if they're feasible (people throw around the number 10^500).
Now, it turns out that the real problem is not actually in the number of distinct solutions to string theory (~10^500), but rather in the way their structure is poorly understood. In fact, any theory of physics contains an infinite number of theories within it. For example, consider the mass of the electron as a free parameter. Until I tell you what the mass is, you can't make a complete prediction for what the energy levels in Hydrogen are. In fact, you could argue that since the mass of the electron is a real number, there are in fact an infinite number of predictions to the energy levels. A bit sarcastically you could then say that at least string theory has "only" 10^500 different theories within it, unlike traditional physics that has this continuous infinite set of theories.
The distinction between these two cases is then that for traditional theories, we can go the other way. If we measure the energy levels of Hydrogen, we can infer the mass of the electron. Then, knowing the mass of the electron, we can make other predictions. It's this last step that's currently missing in string theory. Currently we only know how to move in one direction: give me the solution you're talking about and I might be able to make predictions, but give me observational data and I can't work backwards to determine which solution I'm in. It's almost like a one-way hash.
I would say that this last objection is a roadblock that we're currently facing, and it's not perfectly clear that it's not solvable, nor is it clear that it is. I think that until we solve this problem, string theory will be stuck and people will be pointing fingers at the theory calling it a fool's errand. I personally think this criticism is misguided.
The upshot of this is that most people who work on string theory work in areas that are not plagued by this bifurcation to low energies. For example, you can use string theory to study the structure and behavior of black holes and holography, something called AdS/CFT, an area that has been incredibly successful.