Past research efforts have developed several self-centering beam-to-column connections for minimizing structural damage in steel moment frames after severe earthquakes. However, less focus has been on low-damage column base connections. Severe damage at the base of columns in conventional steel structures with moment frames is likely during an intense earthquake, rendering the building nonresilient. This paper evaluates the application of a recently developed viscoplastic damper in low-damage column base connections. The proposed viscoplastic damper is a rubber-steel core damper (RSCD), which consists of a high-damping rubber layer and several ductile steel bolts. The proposed column base connections prevent damage and inelastic deformations in the steel column and its base plate by allowing a rocking mechanism at the base of the column and limiting damage to easily replaceable steel bolts in the dampers. The paper discusses the design and behavior of the column base connections equipped with RSCDs. Continuum finite element models of RSCD dampers were developed and validated using experimental results. The rocking response was validated using experimental results for rocking columns with a silt damper. A total of 21 finite element models of rocking columns with RSCD were used to numerically examine the column response to combined axial loading and cyclic lateral loading. The influence of different parameters on the cyclic response of the proposed column base connections was also evaluated. These parameters are steel bolt diameter-to-length ratio (d=h), rubber thickness, column axial compressive load, number of bolts, and steel material type. The results confirmed the effectiveness of the proposed column base connection for minimizing damage to the steel column and its base plate. An optimal d=h ratio of 0.4 was found for the design of RSCDs. In addition, analytical formulas are presented to evaluate the yield and ultimate strength of the proposed column base connection, and the comparison with FEM results indicates that the presented mechanism has sufficient accuracy.