What you want to do is called a Spatial Join.
A common GIS task is to join the attributes from one spatial data
layer to another. In this example we will join the attribute table
from a polygon layer to a points layer, based on which polygon
contains the points.
In QGIS, you can do that by using the JOIN ATTRIBUTES BY LOCATION TOOL.
You can then set your target and join vector layers.
I suggest that you set the river basins as the target vector layer and countries as the join layer. That way, your output would be a basins shapefile with the attributes of the countries shapefile. You could then generate your list from there.