Hi all, :D
Ideally you want to wait until the new growth has hardened a bit, ie, wait for the green stems to turn brown. Take a cutting about 10 inches long, remove the leaves & push it into the ground so that 4 to 6 buds are below ground, blackcurrants send shoots up from below ground, so several should come up if you do this. Place 6 to 8 of these cuttings in a row about 6 inches apart, they should grow well & next Autumn they should be ready to transplant to their final growing spot.
Adrian.