By definition, a line only has length. Using linear geometry to model rivers, streams, and canals is a common practice but clearly has it's limitations. If the data exists, you could model your streams using a linear reference system approach where reaches are given a width. Very data intensive. You could also buffer the lines with some sort of average width and find area that way. Just to toss another challenge at you, shouldn't channel depth and or morphometry be considered as well?
That should just about do it....