::waves at all::
You sure can put up the router and a couple of APs in the house. I actually have 2 routers going--one just is in "AP" mode as per the manual for it.
I've read good reviews on the netgear powerline AP system (google for: NETGEAR XE104 or NETGEAR WGXB102) but as always, do your own research--several companies make similar, competing products.
Couple things I'd ask though. Is any part of your system wired? Meaning (as mentioned in the previous post) any chance at moving that router to a different point in the house? (if it's cable or DSL, then nearly any outlet in the house should work)
Anyway to move the current router around in the basement? up high? closer to the stairs?
If you do consider picking up another router or 2 for your APs, check the manual closely as to setting it up for AP use as opposed to full on Wireless router use. Either way, you'll also have to do some admin work on it as the default settings will cause conflicts with the existing router. It's not too hard if you're comfy with IP addresses. I've found a picture will often work well to keep my head straight about what I'm trying to do--even for just a couple of units.
good luck!