RivEX can take a layer and for each feature identifies which bank side the feature is on. This tool will accept point/line/polygon geometry as input. More advice on this tool is found in the manual.

Identify bank side

Linear bank features showing their vertices. Vertices are used to determine which bank side a feature is on, using a "majority wins" approach. In this example you see the Embankment has a single erroneous vertex on the opposite bank but RivEX correctly tags the asset as being on Left bank.