harvest the fruit first then cut back any twiggy growth that will not produce any or much fruit. also cut back any older or heavy branches.

With red currants the important thing to remember when pruning is that they bear fruit on the old wood, the opposite of black currants. So I would not cut out the older branches except any unhealthy ones or any that cross over. Cutting off a lot of this years growth will be OK. Six feet is really taller than they need to be and it will be fine to reduce the height. I'd wait until early autumn to do it. If you stick some prunings into the ground most should root and produce new bushes to give away to your friends. (Don't try rooting the old thick main branches though)
Have you been feeding it with a heavy nitrogen feed? If so give over! You want to encourage flowers and for that you need potash. Try feeding with sulphate of potash in late winter.

Redcurrants do very well as wall trained cordons so you can see that most new growth can be disposed of once you have the framework as you want it.

And don't worry about the stem colour - the stems are naturally very dark.