Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams are very sensitive to perturbations in the input space. The central objective of our work is to develop outlier robust persistence diagrams for topological data analysis. We do so by employing robust density estimators constructed using reproducing kernels. Using an analogue of the influence function on the space of persistence diagrams, we investigate the sensitivity of persistence diagrams from filtrations induced by several functions (e.g., KDE, robust KDE, DTM) and demonstrate the robustness of the proposed method. The robust persistence diagrams are also shown to be consistent estimators in bottleneck distance, with the convergence rate controlled by the smoothness of the kernel, i.e., under some appropriate choices, it is possible to construct outlier-robust persistence diagrams without compromising on statistical efficiency. Finally, we demonstrate the performance of the proposed approach in machine learning applications.