Modern scientific applications frequently produce data that can be modeled by multivariate linear regression. Heavy-tails or outliers often go with them. We propose the nuclear norm penalized multivariate Huber regression to estimate the multivariate mean vector in the high-dimensional setting without presence of symmetry or light tails of the error distribution. We establish a risk bound of the estimator in Frobenius norm under some mild conditions, from which we show that, under proper choice of the Huber constant, the estimator is consistent in estimating the multivariate mean vector. We use the multi-response accelerated proximal gradient algorithm, which is globally convergent, to solve the regression problem. To show the effectiveness of our method, we compare the nuclear norm penalized multivariate Huber estimator with other regularized estimators based on the least squares. Numerical studies for the normal and t-errors are dedicated to show the accuracy of the proposed method. We also apply the multivariate Huber regression model to make a prediction for a real data set. The results demonstrate the good performance of the estimator in terms of the prediction error and its robustness to heavy-tailed errors.