The effect of axisymmetric dipolar fields which are either restricted to the stellar crust, “crustal fields”, or allowed to penetrate the core, “core fields”, on the temperature distribution in neutron star crusts is investigated. While core fields result in practically isothermal crusts unless the surface field exceeds 10¹⁵ G, crustal fields with surface strength above a few times 10¹² G cause significant deviations from crustal isothermality if the core temperatures is ≤ 10⁸ K. At the stellar surface, the cold equatorial region produced by the suppression of heat transport perpendicular to the field by the Larmor rotation of the electrons in the envelope, present for both core and crustal fields, is significantly extended by that classical suppression at higher densities in the case of crustal fields. Then, two small warm polar regions appear which have observational consequences: the neutron star has a small effective thermally emitting area and the X-ray pulse profiles are expected to have a distinctively different shape compared to the case of a core field. These features, when compared with X-ray data on thermal emission of young cooling neutron stars, provide a first step toward a new way to study the magnetic flux distribution within a neutron star.