I know it's possible, but it might require using something other than a line tool to do it. As far as I can recall, there's no way to create a taper on a stroked line.

Your options, then, are to make the rivers with a shape, so that you can explicitly control the width at every point, or to add a small shape at the source of the river that creates the taper.

The first approach is most flexible, but also more work.

There might also be a sheet effect that would fade out the ends of the line. You'd have to run the rivers a little longer than you intended and then mask them, maybe by putting the ocean sheet on top so that the rivers actually run underneath the sea. The fade out wouldn't make them taper at the head, but it might at least ameliorate the problem that they just appear at full size from nowhere.

I think there is also a way to attach an arrowhead to a line. If you make an arrowhead that is just a triangle with the same width as the line instead of having the little "wings" that are wider than the line, that could also work.

It's been a very long time since I last opened CC3, so I might be misremembering some things.