A regularization of Wasserstein barycenters for random measures supported on $R^$d is introduced via convex penalization. The existence and uniqueness of such barycenters is proved for a large class of penalization functions. A stability result of regularized barycenters in terms of Bregman distance associated to the penalization term is also given. This allows to compare the case of data made of n probability measures with the more realistic setting where we have only access to a dataset of random variables sampled from unknown distributions. We also analyze the convergence of the regularized empirical barycenter of a set of n iid random probability measures towards its population counterpart, and we discuss its rate of convergence. This approach is shown to be appropriate for the statistical analysis of discrete or absolutely continuous random measures. In this setting, we propose efficient algorithms for the computation of penalized Wasserstein barycenters. This approach is finally illustrated with simulated and real data sets.