The usual argument is that negative energy states are inherently unstable; if energy states are not bounded from below, a negative energy state can always become more negative, emitting positive energy radiation continuously. It turns out, this is more or less what it is believed that happened in the inflationary era:
1) an accelerated expanding cosmos
2) all the positive energy matter we see today.
So negative energy states are only "bad" (or let say just wildly inconvenient) in our currently asymptotically flat space-time, but they probably existed at the very beginning in vast quantities. They probably marginally exist still today in the form of dark energy.
However, i am confused why people extrapolate the idea that states will always try to decay to lower energy states (even if already negative): What happens at a more fundamental level is that systems try to achieve equilibrium by spreading energy evenly across degrees of freedom of all fields. Entropy is nothing but a logarithm in the number of available states reachable for a degree of freedom at a given, well defined energy. This entropy has a minima at zero energy, not at $- \infty$, as would be implied by the common lore. So it is not unreasonable to expect that, negative energy systems would decay to higher energy states, toward the zero energy states that we associate with the vacuum.