Beamforming (BF) training is crucial to establishing reliable millimeter-wave communication connections between stations (STAs) and an access point. In IEEE 802.11ad BF training protocol, all STAs contend for limited BF training opportunities, i.e., associated BF training (A-BFT) slots, which results in severe collisions and significant BF training latency, especially in dense user scenarios. In this paper, we first develop an analytical model to evaluate the BF training protocol performance. Our analytical model accounts for various protocol components, including user density, the number of A-BFT slots, and protocol parameters, i.e., retry limit and contention window size. We then derive the average successful BF training probability, the BF training efficiency and latency. Since the derived BF training efficiency is an implicit function, to reveal the relationship between system parameters and BF training performance, we also derive an approximate expression of BF training efficiency. Theoretical analysis indicates that the BF training efficiency degrades drastically in dense user scenarios. To address this issue, we propose an enhancement scheme which adaptively adjusts the protocol parameters in tune with user density, to improve the BF training performance in dense user scenarios. Extensive simulations are carried out to validate the accuracy of the developed analytical model. In addition, simulation results show that the proposed enhancement scheme can improve the BF training efficiency by 35% in dense user scenarios.