How can I keep a router aligned correctly while running it along the narrow edge of a board?
I'm not sure if I really understand the question, but I think you are trying to shape the edge of a board by running the router against the narrow edge instead of the wide top/bottom. If this is the case, there are a couple of ways to keep the router steady and perpendicular to the edge.  The best way is to mount the router in a table and run your stock against the guide/fence.  If you don't have a router table, you can clamp some larger stock like a 2X4 to the board to make a wider edge for the router to slide on. This will help keep the router platform flat to the work.