The quasiparticle model is extended to investigate the properties of strange quark matter in a strong magnetic field at finite densities. For the density-dependent quark mass, self-consistent thermodynamic treatment is obtained with an additional effective bag parameter, which depends not only on the density but also on the magnetic field strength. The magnetic field makes strange quark matter more stable energetically when the magnetic field strength is less than a critical value of the order 10(7) Gauss depending on the QCD scale Lambda. Instead of being a monotonic function of the density for the QCD scale parameter Lambda > 126 MeV, the effective bag function has a maximum near 0.3-0.4 fm(-3). The influence of the magnetic field and the QCD scale parameter on the stiffness of the equation of state of the magnetized strange quark matter and the possible maximum mass of strange stars are discussed.