Of course, confirmation of a result like this isn’t a discrete matter: It’s not like it’s unconfirmed, and then there’s one more experiment, and click, it’s confirmed. Nor is it all about the quantity of experiments: Quality counts, too.
That said, there are a few things I’d like to see. First, instead of doing the timing at the source and the detector, do the timing at two different detectors, one near and one far: This will cancel out any systematic errors in the detector readout (the MINOS experiment at Fermilab already did this).
Second, put sources on both ends, so you can perform the experiment both from A to B and from B to A. This will cancel out any errors due to clock synchronization.
Third, if this effect is real, then some explanation is necessary for why it didn’t show up in the SN1987a neutrinos. Presumably, there’s an energy dependence, since these neutrinos were orders of magnitude more energetic than the supernova ones. I’d like to see a decent theoretical model for this effect that’s consistent with the two results.
Finally, once you have such a theoretical framework and calibrate it with the supernova results and results like this, it should be able to predict the speed for neutrinos of intermediate energy. Run experiments at those energies to test it. Plus, of course, other experiments to test whatever other predictions those models make.
Actually, it might be easier to put the detector out in space and the emitter here: Neutrino detectors need to be big, but they’re not all that complicated. You might be able to install an Ice Cube-like detector in one of Mars’ polar ice caps more easily than building an accelerator out there.
And a space-based experiment would also have the advantage that you could run a light beam along the same path, to compare the neutrino speed directly to the speed of light.