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This paper presents a method to damp sloshing in a liquid container during its transfer in a cart equipped with an active vibration reducer having a six-degree-of-freedom parallel linkage. Sloshing is generated during the liquid container's transfer. To damp this sloshing, the active vibration reducer tilts and horizontally moves the container. The reducer is controlled by a linear quadratic regulator. The weighting matrix of the quadratic performance index is optimized with a genetic algorithm (GA). The amplitude of sloshing is considered for the fitness of the GA. In this study, the cart was driven along a straight path on a horizontal plane. The usefulness of the control system was verified through a simulation and experiment.