We investigate the properties in finite magnetic field of an extended anisotropic XXZ spin-1/2 model on the kagome lattice, originally introduced by Balents, Fisher, and Girvin [Phys. Rev. B 65, 224412 (2002)]. The magnetization curve displays plateaus at magnetization m = 1/6 and 1/3 when the anisotropy is large. Using low-energy effective constrained models (quantum loop and quantum dimer models), we discuss the nature of the plateau phases, found to be crystals that break discrete rotation and/or translation symmetries. Large-scale quantum Monte Carlo simulations were carried out in particular for the m = 1/6 plateau. We first map out the phase diagram of the effective quantum loop model with an additional loop-loop interaction to find stripe order around the point relevant for the original model as well as a topological Z(2) spin liquid. The existence of a stripe crystalline phase is further evidenced by measuring both standard structure factor and entanglement entropy of the original microscopic model.