Likelihood cross-validation for kernel density estimation is known to be sensitive to extreme observations and heavy-tailed distributions. We propose a robust likelihood-based cross-validation method to select bandwidths in multivariate density estimations. We derive this bandwidth selector within the framework of robust maximum likelihood estimation. This method establishes a smooth transition from likelihood cross-validation for nonextreme observations to least squares cross-validation for extreme observations, thereby combining the efficiency of likelihood cross-validation and the robustness of least-squares cross-validation. We also suggest a simple rule to select the transition threshold. We demonstrate the finite sample performance and practical usefulness of the proposed method via Monte Carlo simulations and a real data application on Chinese air pollution.