The geological carbon storage in basalt formations involves the interaction of injected CO2 with formation fluids and host rock minerals, which may lead to permanent immobilization through mineral trapping. As the efficiency and spatial distribution of mineralization are controlled by the coupled thermal, hydraulic, mechanical, and chemical processes in the subsurface, the reliable assessment of storage performance relies on predictive multiphysics simulation models. In this work, we present a fully coupled Thermo–Hydro–Mechanical–Chemical (THMC) numerical model implemented within the MOOSE (Multiphysics Object-Oriented Simulation Environment) for simulating CO2 injection and mineral trapping in basalt reservoirs. The framework integrates multiphase flow, heat transport, poroelastic deformation, and reactive geochemistry, incorporating detailed mineral transformation kinetics and a dynamic porosity–permeability feedback mechanism to capture the evolving hydraulic properties of the reservoir. We verify the model by comparing the THM coupling response against a 1D semi-analytical solution, while the chemical module is validated through standardized reactive transport benchmarks. Finally, the model is employed to investigate the sequestration potential of a representative basaltic reservoir in the Shanghai region. Numerical results characterize the spatial-temporal evolution of the mineralization front and reveal the extent to which the interplay between structural deformation and chemical precipitation influences the long-term storage capacity. This study elucidates the dynamics of carbon mineralization and provides a robust theoretical foundation for assessing the feasibility of large-scale storage in reactive formations.