In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first design and analyze the Zeroth-Order Gradient Descent Ascent (ZO-GDA) algorithm, and provide improved results compared to existing works, in terms of oracle complexity. Next, we propose the Zeroth-Order Gradient Descent Multi-Step Ascent (ZO-GDMSA) algorithm that significantly improves the oracle complexity of ZO-GDA. We also provide stochastic version of ZO-GDA and ZO-GDMSA to handle stochastic nonconvex minimax problems, and provide oracle complexity results.