The hydrogen-bonded networks of water at the surface of a model DNA molecule are analyzed. At low hydrations, only small water clusters are attached to the DNA surface, whereas, at high hydrations, it is homogeneously covered by a spanning water network. The spanning water network is formed via a percolation transition at an intermediate hydration number of about 15 water molecules per nucleotide, which is very close to the midpoint of polymorphic transitions between A- and B-forms of the double helix. The percolation transition can occur in both A- and B-DNA hydration shells with nearly identical percolation thresholds. However, the mechanism of the percolation transition in A-and B-DNA is qualitatively different in regard to the roles played by the two opposite grooves of the double helix. Free ions can shift the percolation threshold by preventing some water molecules from hydrogen bond networking. The results corroborate the suggested relationship between water percolation and the low hydration polymorphism in DNA.