We study the influence and structure of the magnetic field in the early phases of low-mass star formation using polarization maps of Bok globules at a wavelength of 850micron, obtained with the Submillimeter Common-User Bolometer Array (SCUBA) at the James Clerk Maxwell Telescope (JCMT). We discuss observations of the following sources: CB26 - a small globule with a nearly dispersed dense core and a young and large circumstellar disk, CB54 - a large globule with a massive dense core and a deeply embedded young stellar cluster, and B335, CB230, and CB244 - three nearby, relatively isolated small globules with low-mass protostellar cores. We find strongly aligned polarization vectors in the case of CB26, B335, and CB230, while the vector orientations in the case of CB54 and CB244 are more or less randomly distributed. The degree of polarization, amounting to several percent, was found to decrease toward the center in each source. Assuming dichroic emission by aligned non-spherical grains as the polarization mechanism, where the magnetic field plays a role in the alignment process, we derive magnetic field strengths and structures from the observed polarization patterns. We compare the magnetic field topology with the morphology and outflow directions of the globules. In the Class 0 sources B335, CB230, and CB244, the magnetic field is oriented almost perpendicular to the ouflow direction. In contrast, the inclination between outflow axis and magnetic field direction is much more moderate (36deg) in the more evolved Class I source CB26.