Before we continue, keep in mind the important disclaimer:
All this discussion is valid assuming the standard model is the correct theory all the way up to the Planck scale, which is unlikely.Indeed, while it's very likely that the standard model is an adequate description of physics at the energies probed by the LHC, we have no compelling reasons to assume it works at, say, 100 TeV. On the contrary, we know there should be some new particles somewhere, at least to account for dark matter and the baryon asymmetry in the universe, and those degrees of freedom may well affect the discussion of vacuum stability. But for the time being let's assume there's no new particles beyond the standard model with a significant coupling to the Higgs field.
The stability of our vacuum depends on the sign of the quartic coupling in the λ |H|^4 term in the Higgs potential: for negative λ the potential is unbounded from below and therefore unstable. We know exactly the value of λ at the weak scale: from the Higgs mass 125 GeV and the expectation value 246 GeV it follows that λ = 0.13, positive of course. But panta rhei and λ is no exception. At large values of |H|, the Higgs potential in the standard model is, to a good approximation, given by λ(|H|) |H|^4 where λ(|H|) is the running coupling evaluated at the scale |H|. If Higgs were decoupled from the rest of matter then λ would grow with the energy scale and would eventually explode into a Landau pole. However, the Yukawa couplings of the Higgs boson to fermions provide another contribution to the evolution equations that works toward decreasing λ at large energies. In the standard model the top Yukawa coupling is large, of order 1, while the Higgs self-coupling is moderate, so Yukawa wins.
In the plot showing the evolution of λ in the standard model (borrowed from the latest state-of-the-art paper) one can see that at the scale of about 10 million TeV the Higgs self-coupling becomes negative. That sounds like a catastrophe as it naively means that the Higgs potential is unbounded from below. However, we can reliably use quantum field theory only up to the Planck scale, and one can assume that some unspecified physics near the Planck scale (for example, |H|^6 and higher terms in the potential) restore the boundedness of the Higgs potential. Still, between 10^10 and 10^19 GeV the potential is negative and therefore it has a global minimum at large |H| that is much deeper than the vacuum we live in. As a consequence, the path integral will receive contributions from the field configurations interpolating between the two vacua, leading to a non-zero probability of tunneling into the other vacuum.
Fortunately for us, the tunneling probability is proportional to Exp[-1/λ], and λ gets only slightly negative in the standard model. Thus, no reason to panic, our vacuum is meta-stable, meaning its average lifetime extends beyond December 2012. Nevertheless, there is something intriguing here. We happen to occupy a very special patch of the standard model parameter space. First of all there's the good old hierarchy problem: the mass term of the Higgs field takes a very special (fine-tuned?) value such that we live extremely close to the boundary between the broken (v > 0) and the unbroken (v=0) phases. Now we realized the potential is even more special: the quartic coupling is such that two vacua coexist, one at low |H| of order TeV and the other at large |H| of order the Planck scale. Moreover, not only λ but also it's beta functions is nearly zero near the Planck scale, meaning that λ evolves very slowly at high scales. Who sets these boundary conditions? Is that yet another remarkable coincidence, or is there a physical reason? Something to do with quantum gravity? Something to do with inflation? I think it's fair to say that so far nobody has presented a compelling proposal explaining these boundary conditions satisfied by λ.
Ah, and don't forget the disclaimer:
All this discussion is valid assuming the standard model is the correct theory all the way up to the Planck scale, which is unlikely.