Mature willows don't need much pruning. The tree will heal faster with less disease problems if you remove broken branches and those that rub against each other. If you shorten the branches, always cut a little beyond a bud or twig. Do not allow branches to grow on the bottom of the tree.
You'll want to get those clippers out at the end of winter. Pruning weeping willow in winter allows you to cut the tree when it is idle. It also makes willows in good condition before they start their spring growth. Weeping willow should only be pruned in late winter and early spring, when the tree is in its dormant period.
This is because willows, like most trees, bleed sap if pruned during their active growth period. When the sap is exposed, it attracts insects, which bring with them spores of fungi and bacteria, which can cause disease. Never proceed with any tree work without first checking if there is a Tree Preservation Order (TPO) in effect. The fact that willow is prone to numerous diseases and pests is another reason why a qualified arborist should really take care of the tree on a regular basis.
The tree mostly has upright branches when young; as it grows, the branches bow to the ground, a habit that can be avoided by pruning the tree regularly. Look for future publications that cover the other willow species or you can ask us to write a guide for your particular tree.