The ground-state properties of a system of charged bosons in a quasi-one-dimensional model with a neutralizing background are investigated within the hypernetted-chain approximation. Strong correlation effects drive the system from a homogeneous fluid phase toward a more ordered structure akin to Wigner crystallization in higher-dimensional charged quantum systems. The ordered phase of charged bosons is signaled by the development of a peak in the static structure factor, which is analyzed as a function of the density and the lateral width of the one-dimensional structure. We also calculate the pair-distribution function, the ground-state energy, and the local-field correction, and compare our results with other theoretical approaches.