We study the problem of using i.i.d. samples from an unknown multivariate probability distribution $p$ to estimate the mutual information of $p$. This problem has recently received attention in two settings: (1) where $p$ is assumed to be Gaussian and (2) where $p$ is assumed only to lie in a large nonparametric smoothness class. Estimators proposed for the Gaussian case converge in high dimensions when the Gaussian assumption holds, but are brittle, failing dramatically when $p$ is not Gaussian. Estimators proposed for the nonparametric case fail to converge with realistic sample sizes except in very low dimensions. As a result, there is a lack of robust mutual information estimators for many realistic data. To address this, we propose estimators for mutual information when $p$ is assumed to be a nonparanormal (a.k.a., Gaussian copula) model, a semiparametric compromise between Gaussian and nonparametric extremes. Using theoretical bounds and experiments, we show these estimators strike a practical balance between robustness and scaling with dimensionality.