Low-propulsion vessels can take advantage of powerful ocean currents to navigate towards a destination. Recent results demonstrated that vessels can reach their destination with high probability despite forecast errors. However, these results do not consider the critical aspect of safety of such vessels: because their propulsion is much smaller than the magnitude of surrounding currents, they might end up in currents that inevitably push them into unsafe areas such as shallow waters, garbage patches, and shipping lanes. In this work, we first investigate the risk of stranding for passively floating vessels in the Northeast Pacific. We find that at least 5.04% would strand within 90 days. Next, we encode the unsafe sets as hard constraints into Hamilton-Jacobi Multi-Time Reachability to synthesize a feedback policy that is equivalent to re-planning at each time step at low computational cost. While applying this policy guarantees safe operation when the currents are known, in realistic situations only imperfect forecasts are available. Hence, we demonstrate the safety of our approach empirically with large-scale realistic simulations of a vessel navigating in high-risk regions in the Northeast Pacific. We find that applying our policy closed-loop with daily re-planning as new forecasts become available reduces stranding below 1% despite forecast errors often exceeding the maximal propulsion. Our method significantly improves safety over the baselines and still achieves a timely arrival of the vessel at the destination.