The harmonic dynamics of normal modes of double-stranded DNA in a viscous fluid are investigated. The model DNA consists of two backbone-supported DNA strands coiling around a common helix axis with base stacking, sugar puckering, interstrand hydrogen bonding, and intrastrand sugar-base interactions assigned values based on published data. Assuming that the DNA bases are shielded from direct bombardment by the solvent, analytical solutions are obtained. The dissipation and fluctuation of the normal modes of the bases moving along the spirals display the effect of the medium indirectly through interactions with the backbone. The dynamics of the backbone are found to be overdamped with the characteristic damping times extending to the picosecond region for disturbance in position and to the sub-picosecond region for disturbance in velocity. In addition to the dynamic mode of a rigid rod, the motions of the bases are coupled to the motions of the backbone with comparable amplitudes for disturbance in position. For disturbance in velocity, however, the bases are effectively at rest, not being able to follow the motions of the backbone. The angular frequencies of the underdamped vibrational modes, identified as the ringing modes of the bases with the backbone effectively at rest, are insensitive to the viscosity and lie in the low frequency region of the Raman spectrum. These findings indicate that the backbone of DNA plays a significant role in modulating the dynamics of double-stranded DNA in an overdamping environment. This modulation of the dynamics of the motions of the bases in DNA by environmental impediments to molecular motion is briefly discussed in connection with protein- and drug- DNA interactions as well as gene regulation.