This paper investigates physical-layer key generation (PLKG) in multi-antenna base station systems, by leveraging a fluid antenna system (FAS) to dynamically customize radio environments. Without requiring additional nodes or extensive radio frequency chains, the FAS effectively enables adaptive antenna port selection by exploiting channel spatial correlation to enhance the key generation rate (KGR) at legitimate nodes. To comprehensively evaluate the efficiency of the FAS in PLKG, we propose an FAS-assisted PLKG model that integrates transmit beamforming and sparse port selection under independent and identically distributed and spatially correlated channel models, respectively. Specifically, the PLKG utilizes reciprocal channel probing to derive a closed-form KGR expression based on the mutual information between legitimate channel estimates. Nonconvex optimization problems for these scenarios are formulated to maximize the KGR subject to transmit power constraints and sparse port activation. We propose an iterative algorithm by capitalizing on successive convex approximation and Cauchy-Schwarz inequality to obtain a locally optimal solution. A reweighted $\ell_1$-norm-based algorithm is applied to advocate for the sparse port activation of FAS-assisted PLKG. Furthermore, a low-complexity sliding window-based port selection is proposed to substitute reweighted $\ell_1$-norm method based on Rayleigh-quotient analysis. Simulation results demonstrate that the FAS-PLKG scheme significantly outperforms the FA-PLKG scheme in both independent and spatially correlated environments. The sliding window-based port selection method introduced in this paper has been shown to yield superior KGR, compared to the reweighted $\ell_1$-norm method. It is shown that the FAS achieves higher KGR with fewer RF chains through dynamic sparse port selection.