Cable-Driven Parallel Robots (CDPR) have attracted significant research interest for applications ranging from cable-suspended camera applications to rehabilitation and home assistance devices. Most of the intended applications of CDPR involve direct interaction with humans where safety is a key issue. Accordingly, this paper addresses the safety of CDPRs in proposing a strategy to minimize the consequences of cable failures. The proposed strategy consists of detecting a cable failure and avoiding any consequent motion of the end-effector. This is obtained by generating a wrench that is opposite to the direction of the ongoing motion so that the end-effector can reach a safe position. A general formulation is outlined as well as a specific case study referring to the LAWEX (LARM Wire-driven EXercising device), which has been designed within the AGEWELL project for limb rehabilitation. Real-time calculation is carried out for identifying feasible cable tensions, which generate a motion that provides the desired braking force. Simulations are carried out to prove the feasibility and effectiveness of the strategy outlined here in cases of cable failure.