Falsified predictions:
All string theories require more than 3 spatial dimensions, all on equal footing. (Plenty of workarounds have been proposed, like compactification, to work around this obvious and glaring issue.) This has been falsified time and again, most recently by the LHC not generating microscopic black holes.
Superstring theories predict "superpartners" for all known particles, none have been seen so far. The proposed workarounds center on "fixing" superpartner masses to be too heavy to be observable. Such moving goal posts are a standard approach.
Multiple possible universes with wildly different physical laws, ours being one of the least likely ones (hence the hyper-anthropic Boltzmann Brain discussion)
The Not Even Wrong blog has addressed the issue a number of times. Here is an example.
String theory remains popular mainly out of desperation: no one has any better model at this point. Plus some of its ideas, like holography and duality, ring true, even if they aren't literally true. Additionally, some of the math developed by the string theorists ended up being quite useful for some real-world calculations. None of which justifies taking Boltzmann Brains seriously.
Such moving goal posts are a standard approach.
When you're talking about a theory as insanely vague as string theory, this isn't exactly an independent problem - it's just something that comes along with it being insanely vague. Not that that isn't a problem, but accusing them of moving the goalposts doesn't fit.
Their goalposts were already spread all over the field. We've just discovered that many of them weren't the set that the ref is watching... if any are.
Leading inflationary multiverse theorist Andrei Linde (and others) gives significant enough credence to the Boltzmann brain problem to tweak some of the parameters of his theory (here for example http://arxiv.org/abs/hep-th/0611043). Are there more elegant ways to deal with the issue that LWs are aware of that would not require changing physics to accommodate?