From what I can see in your picture it looks like the browning is primarily at the margins of the leaves. It also appears that the damage is not too extensive. The other leaves look pretty healthy, no yellowing or deformation. I suspect that your dogwood is suffering from some abiotic condition. Leaf scorch is common in woody perennials, and we're seeing a lot of it this year. It doesn't just happen from lack of water. Leaves may scorch from too much water as well. Or they can get scorched from strong sunlight when the leaves are wet. Dig a little around the root zone of the tree to see if that area is soaked. They don't like strong sunlight. Starlight dogwoods are a cross between the Cornus kousa and the Pacific Dogwood (Cornus nuttallii). It is know for its tolerance to drought and resistance to diseases.