DNA-based molecular wires have a huge potential for application in molecular electronics. Their electronic transport properties are however not well elucidated. This is mainly due to the extreme sensitivity of charge propagation in DNA to static and dynamic disorder as well as to environmental effects (counterions, water shell, etc). 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. For this purpose we use a tight-binding model embedded in a bosonic bath consisting of a collection of harmonic oscillators. In the absence of interactions with the bath, a temperature independent gap opens in the electronic spectrum. Upon allowing for electron-bath coupling the gap becomes temperature dependent. We show that a crossover from semiconducting to metallic behavior in the low-voltage region of the I-V characteristics can be achieved in some parameter regions. The reason is the appearance of bath-induced polaronic states within the electronic bandgap.We further show that the temperature dependence of the transmission near the Fermi energy displays an Arrhenius-like behavior in agreement with recent transport experiments.