Why does vibrational frequency influence quantum tunneling of a proton? Furthermore, consideration of the tunneling in the framework of the Wigner theory assumes that the tunneling factor is small. However, the $\ce{O-H}$ frequency is very high ($\gg k_\mathrm{B}T/h$), and the H atom tunneling under a potential energy barrier when proceeding either from a ground or from other excited vibration levels may be significant.


Source: German, E. D.; Sheintuch, M. Kinetics of Catalytic OH Dissociation on Metal Surfaces. J. Phys. Chem. C 2012, 116 (9), 5700–5709. DOI:10.1021/jp2106499.

Why does a higher frequency influence the quantum tunneling effect?
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.