As a general rule, your birch trees should not require pruning. If you choose to prune a birch, there are a few things you should know. Birch trees are one of the trees that will "bleed" sap when cut. This depletes the tree of moisture and sugar so you will need to water more frequently while the bleeding continues.
Birch trees require a specific cut to remove a limb. Each branch has a branch collar, a swelling where the branch attaches to the tree. Cut right outside this swelling, on the branch side of the collar. Do not cut on the tree side of the branch collar as this can cause structural damage to the tree and make infection more likely. Make sure the branch collar remains intact. There is often a black line running down the trunk where a branch collar is attached, making the collar on a birch especially easily to see.
Most trees are pruned in the spring, but birches should not be pruned until the leaves have reached full size in late spring. Pruning earlier will make bleeding more likely. Don't prune in the late summer. While the tree will not bleed in the late summer, the unhealed cuts will start to bleed as soon as the sap starts to rise in the spring.
There is nothing you can do to stop the bleeding of cuts on birch trees. Commercial pruning paints don't work; all you can do is provide extra water and wait for the wound to heal by itself.
Avoid all pruning any time bronze birch borers are present as the open cuts invite the borers, which can kill the tree. Once a tree is infected with the bronze birch borer, there is little you can do to save the tree.