We investigate the effect of magnetic fields on the propagation dynamics and the morphology of overdense, radiatively cooling, supermagnetosonic jets, with the help of fully three-dimensional smoothed particle magnetohydrodynamic simulations. Evaluated for a set of parameters that are mainly suitable for protostellar jets (with density ratios between those of the jet and the ambient medium. eta approximate to 3-10, and ambient Mach number M-a approximate to 24), these simulations are also compared with baseline nonmagnetic and adiabatic calculations. Two initial magnetic field topologies tin approximate equipartition with the gas, beta = p(th)/p(B) similar or equal to 1) are considered: (1) a helical field and (2) a longitudinal field, both of which permeate both the jet and the ambient medium. We find that, after amplification by compression and reorientation in nonparallel shocks at the working surface, the magnetic field that is carried backward with the shocked gas into the cocoon improves the jet collimation relative to the purely hydrodynamic (HD) systems, but this effect is larger in the presence of the helical field. In both magnetic configurations, low-amplitude, approximately equally spaced (lambda approximate to 2 - 4R(j)) internal shocks (which are absent in the HD systems) are produced by magnetohydrodynamic (MHD) Kelvin-Helmholtz reflection pinch modes. The longitudinal field geometry also excites nonaxisymmetric helical modes that cause some beam wiggling. The strength and amount of these modes are, however, reduced (by about 2 times) in the presence of radiative cooling relative to the adiabatic cases. Besides, a large density ratio, eta, between the jet and the ambient medium also reduces, in general, the number of the internal shocks. As a consequence, the weakness of the induced internal shocks makes it doubtful that the magnetic pinches could by themselves produce the bright knots observed in the overdense, radiatively cooling protostellar jets. Magnetic fields may leave also important signatures on the head morphologies of the radiative cooling jets. The amplification of the nonparallel components of the magnetic fields, particularly in the helical field geometry, reduces the postshock compressibility and increases the postshock cooling length. This may lead to stabilization of the cold shell of shocked material that develops at the head against both the Rayleigh-Taylor and global thermal instabilities. As a consequence, the clumps that develop by fragmentation of the shell in the ND jets tend to be depleted in the helical field geometry. The jet immersed in the longitudinal field, on the other hand, still retains the clumps, although they have their densities decreased relative to the I-ID counterparts. As stressed in our previous work, since the fragmented shell structure resembles the knotty pattern commonly observed in HH objects behind the bow shocks of protostellar jets, this result suggests that, as long as (equipartition) magnetic fields are present, they should probably be predominantly longitudinal at the heads of these jets.