We present an experimental proposal to achieve a strong photon blockade by employing electromagnetically induced transparency (EIT) with a single alkaline-earth-metal atom trapped in an optical cavity. In the presence of optical Stark shift, both the second-order correlation function and cavity transmission exhibit asymmetric structures between the red and blue sidebands of the cavity. For a weak control field, the photon quantum statistics for the coherent transparency window (i.e., atomic quasi-dark-state resonance) are insensitive to the Stark shift, which should also be immune to the spontaneous emission of the excited state by taking advantage of the intrinsic dark-state polariton of EIT. Interestingly, by exploiting the interplay between the Stark shift and control field, the strong photon blockade at atomic quasi-dark-state resonance has an optimal second-order correlation function g(2)(0)～10-4 and a high cavity transmission simultaneously. The underlying physical mechanism is ascribed to the Stark shift enhanced spectrum anharmonicity and the EIT hosted strong nonlinearity with loss-insensitive atomic quasi-dark-state resonance, which is essentially different from the conventional proposal with emerging Kerr nonlinearity in cavity-EIT. Our results reveal a new strategy to realize high-quality single photon sources, which could open up a new avenue for engineering nonclassical quantum states in cavity quantum electrodynamics.