  We study a superconducting single-charge transistor, where the coherence of
Cooper pair tunneling is destroyed by the coupling to a tunable dissipative
environment. Sequential tunneling and cotunneling processes are analyzed to
construct the shape of the conductance peaks and their dependence on the
dissipation and temperature. Unexpected features are found due to a cross-over
between two distinct regimes, one `environment-assisted' the other
`environment-dominated'. Several of the predictions have been confirmed by
recent experiments. The model and results apply also to the dynamics of
Josephson junction quantum bits on a conducting ground plane, thus explaining
the influence of dissipation on the coherence.
