Some ideas:
Just to confirm - the linework is definitely going uphill, not due to the line direction being reversed when symbolized? Check the linework versus your rasters.
To convert from Raster to Vector, did you use Stream to Feature (Spatial Analyst)?
ArcGIS Help (10.2, 10.2.1, and 10.2.2) How Stream to Feature works
ArcGIS Help (10.2, 10.2.1, and 10.2.2) Stream to Feature
Also, if they are all uniformly reversed, as a workaround, one can reverse the line directionality.
ArcGIS Help (10.2, 10.2.1, and 10.2.2) Flip Line (Editing)
caveats - requires at least a standard license.
- alters the input dataset instead of making a new output, so back up your data before running.
Chris Donohue, GISP