With most planets and planetary candidates detected in the stellar habitable zone (HZ) being super-Earths and gas giants, rather than Earth-like planets, we naturally wonder if their moons could be habitable. The first detection of such an exomoon has now become feasible, and due to observational biases it will be at least twice as massive as Mars. But formation models predict moons can hardly be as massive as Earth. Hence, a giant planet's magnetosphere could be the only possibility for such a moon to be shielded from cosmic and stellar high-energy radiation. Yet, the planetary radiation belt could also have detrimental effects on exomoon habitability. We here synthesize models for the evolution of the magnetic environment of giant planets with thresholds from the runaway greenhouse (RG) effect to assess the habitability of exomoons. For modest eccentricities, we find that satellites around Neptune-sized planets in the center of the HZ around K dwarf stars will either be in an RG state and not be habitable, or they will be in wide orbits where they will not be affected by the planetary magnetosphere. Saturn-like planets have stronger fields, and Jupiter-like planets could coat close-in habitable moons soon after formation. Moons at distances between about 5 and 20 planetary radii from a giant planet can be habitable from an illumination and tidal heating point of view, but still the planetary magnetosphere would critically influence their habitability.