The signature convention $(−,+,\cdots,+)$ is more commonly used in General Relativity and Lorentzian geometry because of the desire among their practicioners to make a closer parallel to Riemannian geometry. This is *not* the case in the remainder of Mathematical Physics (particularly in PDE's and field theory) precisely because of issues such as this. So my guess is, you usually do not find it in papers in the latter areas because there usually one adopts the $(+,−,\cdots,−)$ signature convention and then it becomes a *non*-issue. The latter choice also simplifies matters regarding spinors in Lorentzian manifolds, but that is another story...

That being said, if you look e.g. at Chapter 3 of the book of J. K. Beem, P. E. Ehrlich and K. L. Easley, *Global Lorentzian Geometry* (second edition, CRC Press, 1996), there one can infer the following rationale for choosing the OP's *second* convention (as suggested by the OP itself and Igor in the comments above) if the signature convention for our Lorentzian manifold $(M,g)$ happens to be $(-,+,\cdots,+)$ - that is, $\xi\in T^*\!M$ is a future directed timelike covector if $g^{-1}(\xi,\xi)<0$ (here $g^{-1}$ is the Lorentzian metric on $T^*\!M$ induced by $g$) *and* $\xi^\sharp=g^\sharp(\xi)=g^{-1}(\xi,\cdot)$ is *past* directed. The rationale is the following: consider a (real-valued) smooth function $t$ on $M$ such that $dt$ is *timelike*. If $t$ is a *global time function* on $(M,g)$, that is, a (continuous real-valued) function $t$ on $M$ that *strictly increases* along any *future directed* causal curve on $(M,g)$ (see pp. 64 of *ibid.*), this means that $T(t)=dt(T)=g(dt^\sharp,T)>0$ for any future directed causal vector field $T$, which amounts to saying that $dt^\sharp$ is *past directed*. 

In other words, if we are to understand a global time function on $(M,g)$ as a global time coordinate that grows from the past into the future according to our choice of time orientation on $(M,g)$, it is much more natural to set the timelike covector field $dt$ as *future directed*. This also seems to be the reasoning adopted in R. M. Wald's *General Relativity* (University of Chicago Press, 1984), see e.g. Theorem 8.2.2, pp. 198-199 therein and the discussion that precedes it. Since these two books are quite often used as references in their field, it seems reasonable to me to reckon the above choice for the time orientation on $T^*\!M$ as the most common (albeit not the only) one with the signature convention $(-,+,\cdots,+)$.

If one applies this reasoning to the signature convention $(+,-,\cdots,-)$, one concludes instead that if a (smooth) global time function $t$ on $(M,g)$ has a timelike differential $dt$ as in the above paragraph, then $dt^\sharp$ becomes a *future directed* timelike vector field on $(M,g)$. This, of course, is consistent with the (resp. absence of) "sign flipping" of timelike (co)vectors when "raising and lowering indices" (that is, respectively applying the musical isomorphisms $g^\sharp$ and $g^\flat=(g^\sharp)^{-1}$) with the signature convention $(-,+,\cdots,+)$ (resp. $(+,-,\cdots,-)$). Using time functions (which, more generally, always exist *locally* on $(M,g)$) to set the time orientation on $T^*\!M$ is then also seen to have the advantage of being independent of the signature convention on $(M,g)$.