Recent advances in Autonomous Underwater Vehicle (AUV) technology have facilitated the collection of oceanographic data at a fraction of the cost of ship-based sampling methods. Unlike oceanographic data collection in the deep ocean, operation of AUVs in coastal regions exposes them to the risk of collision with ships and land. Such concerns are particularly prominent for slow-moving AUVs since ocean current magnitudes are often strong enough to alter the planned path significantly. Prior work using predictive ocean currents relies upon deterministic outcomes, which do not account for the uncertainty in the ocean current predictions themselves. To improve the safety and reliability of AUV operation in coastal regions, we introduce two stochastic planners: (a) a Minimum Expected Risk planner and (b) a risk-aware Markov Decision Process, both of which have the ability to utilize ocean current predictions probabilistically. We report results from extensive simulation studies in realistic ocean current fields obtained from widely used regional ocean models. Our simulations show that the proposed planners have lower collision risk than state-of-the-art methods. We present additional results from field experiments where ocean current predictions were used to plan the paths of two Slocum gliders. Field trials indicate the practical usefulness of our techniques over long-term deployments, showing them to be ideal for AUV operations.