The fundamental challenge of the millimeter-wave (mmWave) frequency band is the sensitivity of the radio channel to blockages, which gives rise to unstable connectivity and impacts the reliability of a system. To this end, multi-point connectivity is a promising approach for ensuring the desired rate and reliability requirements. A robust beamformer design is proposed to improve the communication reliability by exploiting the spatial macro-diversity and a pessimistic estimate of rates over potential link blockage combinations. Specifically, we provide a blockage-aware algorithm for the weighted sum-rate maximization (WSRM) problem with parallel beamformer processing across distributed remote radio units (RRUs). Combinations of non-convex and coupled constraints are handled via successive convex approximation (SCA) framework, which admits a closed-form solution for each SCA step, by solving a system of Karush-Kuhn-Tucker (KKT) optimality conditions. Unlike the conventional coordinated multi-point (CoMP) schemes, the proposed blockage-aware beamformer design has, per-iteration, computational complexity in the order of RRU antennas instead of system-wide joint transmit antennas. This leads to a practical and computationally efficient implementation that is scalable to any arbitrary multi-point configuration. In the presence of random blockages, the proposed schemes are shown to significantly outperform baseline scenarios and result in reliable mmWave communication.