For simplicity, let's consider the case of holomorphic modular forms over $\mathbb Q$ of squarefree level and trivial nebentypus. Then one knows from

Iwaniec, Henryk; Luo, Wenzhi; Sarnak, Peter. Low lying zeros of families of $L$-functions. Publications Mathématiques de l'IHÉS, Tome 91 (2000) pp. 55-131.

an asymptotic formula for the dimensions of the subspaces of cusp forms with root number $+1$ and root number $-1$. In particular, the proportion of forms with root number $+1$ tends to $\frac 12$ as you take some combination of the weight and the level to infinity.

As Peter mentions in the comments, I also consider this in my paper

Refined dimensions of cusp forms, and equidistribution and bias of signs. Journal of Number Theory, Vol. 188 (2018), 1-17.

Specifically, I get an exact formula for the dimensions of subspaces with prescribed root number (or prescribed Atkin-Lehner signs), and observe a "strict bias" phenomenon for root number +1: while the proportion of newforms with root number +1 is asymptotically $\frac 12$, in any given space, there are always at least as many newforms with root number +1 as with -1, and it is strictly greater except in a few special situations.

One should similarly be able to prove that the proportion is $\frac 12$ in more general families of automorphic forms, say using a trace formula and simply bounding error terms, though I don't know the most general situation in which this has already been done in the literature. Essentially one needs to know that the trace of the Fricke involution is not large compared to the dimension.