The bronze birch borer is rarely a problem on river birch so I doubt this is the cause of the decline.  The most common reason for discoloration of the foliage and tree dieback is chlorosis, due to the inability of the tree to absorb sufficient iron or manganese.  The underlying problem (no pun intended) is the alkalinity of the soil reduces the amount of soluble FE and MN so merely adding it to the soil will not usually help.  The best solution may be to find a chelated form of these elements at a local garden center and apply this fertilizer following label directions.  Do not expect a quick return to health, it may take a couple of years of annual applications.