Can running water carry an electrical current indefinitely, and for long distances?
I assume that the fundamental problem with this idea is that due to inherent resistance of the water, heat will be generated. As a result the water will evaporate. If there is no space left for the gaseous water vapor the "pipeline" will just explode.

That is basically why conducting wires are made of copper, it mirrors a remarkably low resistance. Silver and gold would be even better but more expensive on the other hand. Consequently copper is the standard material of wires.

If constant cooling of the system would be permitted, it could basically work. 
water itself does not significantly conduct electricity, but with electrolytes dissolved in the mentioned (as suggested by the author of this question) it becomes conducting.
As long as the ions do not precipitate the liquid retains the property of conductance.