Why does vibrational frequency influence quantum tunneling of a proton?
It is not the frequency per se but the mass that is important,; the probability of tunnelling an energy E is given by  

$$p(E)=\exp \left(-\frac{4\pi\sqrt{m}}{h^2}\int_{x_a}^{x_b}\sqrt{V(x)-E}\;dx \right)$$

where $x_a$ and $x_b$ are two points either side of the potential barrier through which tunnelling occurs. The integral is in effect a measure of the area of the potential $V$ above position $x_a$ to $x_b$ through which tunnelling can occur.

As m increases, for a given potential barrier and energy, the tunnelling probability falls.