The important thing to remember is that this is a grafted tree, i.e. the 'weeping' part at the top has been grafted on to another Salix rootstock. If you see any growth at all below the graft this needs to be removed as it will be far more vigorous than the weeping part and totally spoil the look of the tree.

As to actual pruning please don't give it a 'pudding basin' haircut, thin it out by removing a few selected branches, right back to the centre. This reduces congestion, and retains the shape.

If you have one that is a dense mass, get down on hands and knees and work your way under the umbrella of foliage to the central trunk. Reach upwards, and start by cutting out all the dead stuff you can see - early spring is a good time to do this as the live shoots will be budding, making it easy to spot the dead ones. You may still need to remove more branches if it is very congested - again work from inside so that the pruning is not visible from the outside of the tree.

Ours overhangs a stone dyke and can go right down to the road, we then have to remove branches to prevent it wandering across said road, mind you that could work as a traffic calming measure, lol!

I feel so much calmer now that the neighborhood has put in a "traffic calming" island on my route to work! :/
Pruning weeping plants takes a bit of skill and thought, as MG says. Fortunately, mistakes made in the learning are at least partially self-correcting, over time. So, don't stress! :)