Sonds like the wood isn't really a quality wood and while you plan on putting a runner on over it, to hide it's major "defects", it will probably NEVER look as nice as a finished hardwood or heart-of pine.
A paint wash MIGHT work --- have you got some "spare" wood that you can test the two approaches (stain plus polyurethane; white-wash plus polyurethane)? And it isn't clear to me: are you white washing before or after staining? Or white washing only? There could be a reaction between the white wash and the stain is why I asked...
Also, I know that water will make grain on some woods "rise" which could make it MORE obvious... which means a test is even more important since grain is NOT what you want.
Another option: have you considered painting with a faux woodgrain finish?
here's a really good article on how to refinish wooden stair treads:
http://www.associatedcontent.com/article/841481/how_to_refinish_wood_stair_treads.html?cat=6
Hope this helps...