Do states have any special responsibilities or constraints on their management of beds and banks of navigable rivers?

Yes. Regardless of whether a river is designated as a wild and scenic river, states have special responsibilities and management constraints with respect to state-owned lands underlying navigable waters. These special responsibilities arise from the Public Trust Doctrine, which requires states to exercise regulatory authority over navigable riverbeds to ensure that the paramount right of public use of the rivers and riverbeds for navigation, commerce, recreation, and related purposes is not substantially impaired. As a matter of common law, the states hold lands underlying navigable waters in trust for the benefit of the public. State responsibility may be delegated by the state to local governments.

Source
Interagency Wild & Scenic Rivers Council