I currently have an ASUS RT-N66U Dark Knight Double 450 mbps N Router in my basement, but I have thin wi-fi coverage upstairs. Should I upgrade to a better router in the basement and use the existing router as an extended upstairs?
I would use a wifi extender rather than replace the basement router. You can use a wifi router as an extender but if the setting get changed back to default (like with power outage) it will revert to a full router and that will cause problems.
If you use a wireless extender, you will have slower wireless speeds on the far connection. What I have done is use a powerline connection to setup an ethernet jack at the far end and then use a router setup like an access point or a dedicated access point for wireless.