In principle, yes, but it's observation instead of experiments [at least for now]. Stringy effects at high energy could have contributed in the very early universe, and with the expansion of the universe even tiny effects can be magnified. In the current non-stringy standard cosmology, you can use quantum field theory to study what is the effect of quantum fluctuation to matter perturbation, and this is basically the seeds that later form galaxies. In certain models of string theory, stringy effects will modify this, and cosmological observations can in principle falsify some of these models.There may also be imprints on the cosmic microwave background that differ from non-stringy models [http://arxiv.org/abs/0704.0647v1]
. Of course all these are works and ideas in progress, but in principle they can falsify or at least put some constraints on string theory.
See also http://www.math.columbia.edu/~woit/testable.pdf