There *is* a version of the Chebotarev density theorem for finitely generated fields, or more precisely, after spreading out, for an étale Galois cover of schemes of finite type over a ring of $S$-integers. This is a consequence of work surrounding the Weil conjectures. See Lemma 1.2 in Torsten Ekedahl, An effective version of Hilbert's irreducibility theorem, Séminaire de Théorie des Nombres, Paris 1988-89, Birkhäuser 1990.

But what I would recommend trying first is to use the monodromy groups. If you spread out your Galois extension to a finite branched cover of a dense open subscheme $U$ of $\mathbf{P}^n_{\mathbf{Q}}$, then the monodromy group (inertia group) associated to any irreducible divisor on $U$ is contained in your Galois group. In particular, if there is a divisor that ramifies in the simplest possible way (ramification indices $2,1,1,\ldots,1$), then you know that your group contains a transposition.

The other approach to try, already mentioned by others in the comments, is specialization, which amounts to spreading out as above and then restricting to the cover above an irreducible closed subscheme. More concretely, you could plug in rational numbers for some or all of $y_1,\ldots,y_n$ such that you get a separable polynomial (of the same degree as the original polynomial). Or you could spread out to a scheme of finite type over $\mathbf{Z}$ and then restrict to an irreducible closed subscheme; this includes reduction modulo primes. If specialization results in a separable polynomial of the same degree over the new function field, then the Galois group of the specialization is a subgroup of the original Galois group, so it suffices to show that the specialized Galois group is not contained in $A_n$.

For more techniques for computing Galois groups, I recommend the book *Topics in Galois theory* by Jean-Pierre Serre.