We study the order parameter phase fluctuation effects in cuprate superconductors near T = 0, using a quasi-two-dimensional d-wave BCS model. An effective phason theory is obtained which is used to estimate the strength of the fluctuations, the fluctuation correction to the in-plane penetration depth, and the pair-field susceptibility. We find that, while the phase fluctuation effects are difficult to observe in the renormalization of the superfluid phase stiffness, they may be observed in a pair tunneling experiment which measures the pair-field susceptibility.