This paper concentrates on the random vibration suppression of a regular straight beam by using an inerter-based dynamic vibration absorber. For a wideband random point-driven straight beam with an inerter-based dynamic vibration absorber, the distribution of mean-square velocity response along the axis of the straight beam as well as the mean kinetic energy of the whole beam are first analytically derived through the classical linear random vibration theory. Two optimization objectives are established to determine the optimal design parameters: (1) minimizing the maximal mean-square velocity along the axis of the straight beam, which corresponds to the maximal mean kinetic energy density along the axis and (2) minimizing the mean kinetic energy of the whole beam. Numerical search gives the optimal location and the associated optimal parameters of the inerter-based dynamic vibration absorber. Numerical results for a simply supported straight beam illustrate the better performance of an inerter-based dynamic vibration absorber than a traditional dynamic vibration absorber. Parametric sensitivity studies for the robustness analysis of the beam response to deviations from the optimal parameters are conducted. The optimal location locates on the force-excited point, while the suboptimal location locates on its symmetry position. Furthermore, the optimal and suboptimal locations remain invariable regardless of the upper cutoff frequency of band-limited noise, which is fairly important to the location optimization of the inerter-based dynamic vibration absorber.