We investigate the influence of a dissipative environment which effectively comprises the effects of counterions and hydration shells, on the transport properties of short DNA wires. Their electronic structure is captured by a tight-binding model which is embedded in a bath consisting of a collection of harmonic oscillators. Without coupling to the bath a temperature independent gap opens in the electronic spectrum. Upon allowing for electron-bath interaction the gap becomes temperature dependent. It increases with temperature in the weak-coupling limit to the bath degrees of freedom. In the strong-coupling regime a bath-induced pseudogap is formed. As a result, a crossover from tunneling to activated behavior in the low-voltage region of the I-V characteristics is observed with increasing temperature. The temperature dependence of the transmission near the Fermi energy, t(E-F), manifests an Arrhenius-type behavior in agreement with recent transport experiments. Moreover, t(E-F) shows a weak exponential dependence on the wire length, typical of strong incoherent transport. Disorder effects smear the electronic bands, but do not appreciably affect the pseudogap formation.