The proportion is 1. In fact, there is a nonempty Zariski open set such that all polynomials in the open set satisfy the condition.

Indeed, a sufficient condition, for a polynomial of degree $n$ is that the monodromy group of the covering $\mathbb C \to \mathbb C$ induced by $f$ is the full symmetric group $S_n$. This is sufficient because the covering induced by $f$ mod $p$ will have the same geometric monodromy group for all but finitely many $p$, and will then fail to be injective for all sufficiently large $p$ by Deligne.

A classical sufficient for a polynomial to have monodromy group $S_n$ is that it has $n-1$ distinct critical values (over $\mathbb C$). This is certainly a Zariski open condition, and is satisfied for the polynomial $x^n-x$, hence nonempty.

The proportion is 1. In fact, there is a nonempty Zariski open set such that all polynomials in the open set satisfy the condition.

Indeed, a sufficient condition, for a polynomial of degree $n$ is that the monodromy group of the covering $\mathbb C \to \mathbb C$ induced by $f$ is the full symmetric group $S_n$. This is sufficient because the covering induced by $f$ mod $p$ will have the same geometric monodromy group for all but finitely many $p$, and will then fail to be injective for all sufficiently large $p$ by Deligne.

A classical sufficient condition for a polynomial to have monodromy group $S_n$ is that it has $n-1$ distinct critical values (over $\mathbb C$). This is certainly a Zariski open condition, and is satisfied for the polynomial $x^n-x$, hence nonempty.