The viscosity of water creates drag on swimmer's body so its effect is to slow down the swimmer. However the viscosity seems to be essential for pushing the water backwards by the swimmer's arms and legs. Would a human be able to swim in water with much lower viscosity? What standard stroke (front crawl, breaststroke, butterfly) would work best/worst? How would a lower viscosity affect fish motion in water?
There are two forces you need to consider when calculating fluid flow, viscous and inertial. The former comes from the viscosity of the fluid and the latter from the momentum of the fluid - momentum is mass (of a fluid element) times velocity just as for a particle. In general at very low shear rates viscous forces dominate while at high shear rates inertial forces dominate.
I would guess that when swimming (at least high speed swimming) the thrust you can generate is dominated by inertial forces. When you swim you are in effect throwing water backwards so conservation of momentum propels you forwards. Therefore you would be able to swim even in fluids with very low viscosity. The drag is probably dominated by viscosity, and this would be reduced in a thin liquid, so you might even be able to swim faster as the viscosity is reduced.
Calculating exactly how the efficiency of the various strokes is affected seems hard to me and I'm not sure it would be wise to speculate. It would be easy to do some basic experiments with different fluids and a swimming toy.
Later: it appears Myth Busters have had a play with this and there is an article here. They don't seem to have done any experiments though.