A phenomenological model is introduced for the dissipative quantum dynamics of the phase across a current-biased Josephson junction. The model is invariant under +2. This enables us to restrict to the interval 0 to 2, equating +2 with , and study the role played by the resulting nontrivial topology. Using Feynman's influence functional theory it is shown that the dissipation suppresses interference between paths with different winding numbers. For Ohmic dissipation this interference is completely destroyed, and can effectively be treated as an extended coordinate. This justifies the use of the usual washboard potential description of a current-biased junction even in the quantum case, provided on Ohmic dissipation mechanism is present. © 1986 The American Physical Society.