In the Bay Area, Birch trees are best pruned in Sept/October, right before the leaves fall. Pruning Birches in the Spring and Summer can lead to infestation of the Bronze Birch Borer. Pruning Birches in the Fall and Winter can lead to intense, unwanted suckering in the spring.
If you have a birch tree or would like to plant a birch tree check out this website.
I pruned two Birches in this clients yard. The first tree had to be brought down somewhat because it blocked a neighbor’s view. The second tree simply needed to be cleaned up. When I prune birch trees I like to accent the trailing branches and remove the vertical ones.