Pruning river birch
The river birch is more resistant to insects and diseases than the white birch. If the branches are in your way such as, if you have to mow, you can prune by mid summer. Otherwise, the best time would be in the early fall after the trees shed their leaves or when the leaves emerge in the spring.
Here is our publication on birch. https://extension.umd.edu/sites/extension.umd.edu/files/_docs/programs/hgic/HGIC_Pubs/TreesandShrubs...