"stability" is invoked as the justification for the axiomatic requirement that the spectrum of the generators of the translation group must be confined to the forward light-cone. The spectrum condition has pervasive, significant effects in axiomatic QFT. There seems to be no proof, however, that the spectrum condition actually ensures that a quantum field will be stable, partly because there is, AFAIK, no mathematical specification of what stability consists of in QFT. IF there were, I suppose the stability axiom would be central to axiomatic QFT instead of the positive spectrum condition.
Stability is intimately related with positive energy in classical physics, of course, but the concept of energy is rather different in classical relativistic field physics than in quantum mechanics, being the 00 component of the stress-energy tensor instead of being the 0 component of the 4-vector of generators of translations. The relationship between positive energy and stability in classical physics does not seem enough to justify an uncritical adoption of the spectrum condition in quantum theory as an axiom, which is supposed to be obvious enough that it is almost beyond question. Negative frequencies are certainly not ruled out for classical field theories, because the energy is not a linear functional of the frequency of the Fourier components of the field.
An axiomatic definition of stability would presumably have to specify what deformations would or should not affect the stability of a given construction. A building is only stable, for example, provided a strong enough earthquake does not occur, it is not stable sine die. Given that the deformations that are possible in quantum field theory are more varied than the deformations that are possible in classical field theory, the spectrum condition seems to require a more substantial justification.
Less axiomatically, Feynman integrals include negative frequency/energy components in intermediate calculations, though not in observables, which seems to bring the spectrum condition into at least some question.
Haag discusses the relationship of stability with the spectrum condition only extremely perfunctorily (p.29 of the 2nd edition of Local Quantum Physics), and I am not aware of an elaborate discussion by other authors. Is there one?
EDIT: Streater & Wightman, in PCT, Spin & Statistics, and all that, discuss collision states. Their discussion is entirely in terms of perturbation theory, which seems not adequate enough for an axiomatic discussion. However, because of asking this question I'm starting to see slightly more clearly why a conventional Physicist might be entirely satisfied with what there is on this.
EDIT(after acceptance of Tim van Beek's Answer): The other aspect of this is that the restriction to positive frequency is apparently not enough to ensure “confinement”, at least not in an elementary way. That seems to me to be more what “stability” ought to mean.