On an interface, for normal incident waves, you would use a Cauchy boundary condition, which is both a Neumann and Dirichlet boundary condition. In the one dimension case, this insures there will be a unique continuation and thus a unique full solution. The higher dimensional cases may not be as nice, but usually everything works out.
Now, as per what you added in the comments, the reflection and transmission coefficients are determined by the boundary conditions, but those boundary conditions are not specified independently. They need to match with the specified Helmholtz wave equation.
In the general case, you would suppose continuity and use the wave equation together with any other physical constraints in your model (e.g. in electromagnetism) to determine the exact boundary conditions on the interface. After that, you would solve the Helmholtz equation on the volumes of the different mediums, obtaining some kind of harmonic functions. If separated in cartesian coordinates, then this leads to plane waves. Finally match those different solutions using the boundary conditions as above determined will lead you to the coefficients.
