One never obtains "negative probability" densities when one discusses single observables. One obtains "negative probability" densities only when one discusses joint distributions of incompatible observables, for which the commutator is non-zero (because they take negative values, they are not probability densities). So, to avoid negative probability densities entirely, only discuss joint probability densities of compatible observables.
There are some states in which some pairs of incompatible observables nonetheless result in positive-valued distributions. The best-known examples are coherent states, for which the Wigner function is positive-definite. This, however, does not extend to all possible observables, so that in a coherent state not all pairs of incompatible observables result in positive-definite joint probability densities.
The failure of joint probabilities to exist for all states means that even though positive-definite densities may exist for particular observables in particular states, it is generally taken to be too much to call any positive-definite joint density that might happen in a special class of states to be a probability density just because it is positive-definite.
There is one quite general way to construct an object that is always positive-definite from a Wigner function, which is by averaging it over a large enough region of phase space. Many attempts to do this in a mathematically general way have been constructed over the years. I personally like Paul Busch's approach (with various co-workers), whose web-site lists two monographs that do this quite nicely:
The Quantum Theory of Measurement
Paul Busch, Pekka Lahti, Peter Mittelstaedt. Springer-Verlag, Berlin
Lecture Notes in Physics, Vol. m2, 1991; 2nd ed. 1996
Operational Quantum Physics
Paul Busch, Marian Grabowski, Pekka Lahti. Springer-Verlag, Berlin
Lecture Notes in Physics, Vol. m31, 1995; corr. printing 1997
I'm certain that other people have other preferences, however. For some, this is a way to reconcile quantum with classical, for others it is not.
There is a quick and dirty way of seeing the relationship between incompatibility and positive-definiteness of putatively positive joint probability densities, which can be found in a paper by Leon Cohen, "Rules of Probability in Quantum Mechanics", Foundations of Physics 18, 983(1988). I trot this out quite regularly, even though it's rarely cited in the literature because it's not very nice mathematics, because it's such elementary mathematics and it influenced my understanding of QM a lot a long time ago (I cited it here, for example, for a not very related Question).