How and when sould I prune my Kilmarnock Willow?

The idea of the Kilmarnock Willow is that the branches do go right down to the ground. I suppose that you could trim the ends of the branches if they concern you but it should not be necessary. As it develops the willow will produce a lot of dead twigs under the crown. This is normal and they can be pulled off.
The picture shows a very spendid border; I like it. However, I do not see a Kilmarnock Willow in the picture. The tree in the foreground looks more like a weeping beech?

One prunes all Willows immediately after flowering, so now is a good time. That gives the tree time to produce new growth which will flower next Spring. I prune my dwarf weeping willow back every other year or it gets in an awful tangle.

However, as B points out the pix is not of a willow but of a weeping beech - personally I think it would be a crime to prune...