Is a Betelgeuse supernova able to neutralise earth's nuclear arsenal? According to an article on newscientist.com, a neutrino beam could neutralise nuclear bombs by inducing a slow meltdown of the nuclear fuel. The neutrino generator 

would need to be more than a hundred times more powerful than any existing particle accelerator

Seems not much to me, compared to the "$10^{46}$ joules, approximately 10% of the star's rest mass," which, wikipedia says, "is converted into a ten-second burst of neutrinos, which is the main output of a supernova"
So, is a supernova able to neutralise earth's nuclear arsenal?
 A: No. Ordinary supernovas do not produce neutrinos of large enough energy to cause such a nuclear weapon meltdown, even if the inverse square law diminution of flux is not an issue.
The original paper on which the NewScientist based its article is the preprint

Sugawara, H., Hagura, H., Sanami, T. Destruction of Nuclear Bombs Using Ultra-High Energy Neutrino Beam. arXiv:hep-ph/0305062.

There we could see the principles on which this hypothetical weapon is based upon. To melt a nuclear weapon ultra-high energy neutrino beam (about 1000 TeV, well higher than energy achievable in modern accelerators) is produced. At such energies the mean free path of the neutrino is comparable with radius of the Earth. At the same time the beam would be narrow enough, so its spread at the distance in question is negligible. The high energy neutrinos interacting with matter would produce hadron showers. If such an event happens near the fissionable material, nuclear fission could be initiated by hadrons, and if the power of the beam is large enough, the energy released by such process would be large enough to meltdown a nuclear weapon (cause a fizzle).
However, neutrinos released by an 'ordinary' supernova are of much smaller energies (from several MeV to several dozen MeV). The neutrinos of such an energies do not cause hadron showers (at all), the cross-section of interaction with matter is much smaller, so the induced fission events would be negligible even at a distances where neutrino radiation from supernovae would cause lethal dose (Thanks, Ross Millikan for the reminder). (Incidentally, the ultra-high energy neutrino beam weapon from the paper would also be able to produce lethal dose within seconds). 
So even if supernova event occurs near enough to threaten the life on Earth, neutrinos from it wont damage nuclear weapons. And Betelgeuse is far away enough that had it exploded it wont cause noticeable damage to Earth ecosystem.
One could also note, that ultra-high energy neutrinos detected recently by the IceCube detector, have the energy in that same energy range (>1 PeV), so there probably are astrophysical sources of such neutrinos.
