Prune your black currant bushes in the later winter or early spring when the plants are in their dormancy.
Prune back any branches that lay down on, touch or even barely sweep along the soil. Inspect for branches that are dead, broken, cracked or otherwise ailing. Sever back to the point of healthy, intact wood just 1/4 inch above a branch spur or down to the crown of the plant, if needed.
Deadhead all of the flowers on newly planted bushes in their first year after planting. This will drive plant resources to root and branch development resulting in healthier plants with larger harvests in years hence.
Cut down all but six to eight of the most vigorous shoots on 1-year-old bushes. Cut down all but four or five new shoots and leave three to four 2-year-old canes. In the third year, remove all canes save three to four each of 1-, 2- and 3-year-old canes. Remove the 4-year-old canes in the fourth year and each year thereafter. As a goal, keep eight healthy vigorous fruiting canes on each mature bush at all times.