We analyze the training process of the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight problems arising from essential boundary values. Typically, one employs a penalty approach to enforce essential boundary conditions, however, the naive approach to this problem becomes unstable for large penalizations. A novel method to compensate this problem is proposed, using a small penalization strength to pre-train the model before the main training on the target penalization strength is conducted. We present numerical evidence that the proposed method is beneficial.