Supernova (SN) 2008bk is a well-observed low-luminosity Type II event visually associated with a low-mass red-supergiant progenitor. To model SN 2008bk, we evolve a 12 M_⊙ star from the main sequence until core collapse, when it has a total mass of 9.88 M_⊙, a He-core mass of 3.22 M_⊙ and a radius of 502 R_⊙. We then artificially trigger an explosion that produces 8.29 M_⊙ of ejecta with a total energy of 2.5 × 10^50 erg and ∼0.009 M_⊙ of ^56Ni. We model the subsequent evolution of the ejecta with non-local thermodynamic equilibrium time-dependent radiative transfer. Although somewhat too luminous and energetic, this model reproduces satisfactorily the multiband light curves and multi-epoch spectra of SN 2008bk, confirming the suitability of a low-mass massive star progenitor. As in other low-luminosity SNe II, the structured Hα profile at the end of the plateau phase is probably caused by Ba ii 6496.9 Å rather than asphericity. We discuss the sensitivity of our results to changes in progenitor radius and mass, as well as chemical mixing. A 15 per cent increase in progenitor radius causes a 15 per cent increase in luminosity and a 0.2 mag V-band brightening of the plateau but leaves its length unaffected. An increase in ejecta mass by 10 per cent lengthens the plateau by ∼10 d. Chemical mixing introduces slight changes to the bolometric light curve, limited to the end of the plateau, but has a large impact on colours and spectra at nebular times.