Passive vibration control of an elastic structure carrying a rectangular tank, partially filled with liquid, is investigated when the structure is subjected to horizontal, narrowband, random ground excitation. The modal equations of motion for liquid sloshing are derived using Galerkin's method, considering the nonlinearity of sloshing. The system response statistics including mean square values, correlation coefficients, and probability density functions (PDFs) are numerically estimated from the time histories using the Monte Carlo simulation when the natural frequency of the structure is close to that of liquid sloshing. The influences of the excitation center frequency, its bandwidth, and the liquid level on the system responses are examined. As a result, it is found that the mean square responses of the structure decrease when the center frequency is close to the natural frequency of the structure due to sloshing. Tuned liquid dampers (TLDs) are found to be most effective for comparatively low liquid levels.