You can use rags or paper towels to apply the stain. The stain is wiped off to the degree of color darkness that you are looking for. It is always best to practice on a scrap piece of wood until you achieve the look that you want.
If you are talking about painting the vertical portion of the steps, why? This is the part that people will be kicking as they walk up, thus creating a constant maintenance problem.

Hi Sharon

I'm certainly no expert, but I do enjoy refinishing antiques and wood surfaces in my home.

For staining (non floor surfaces), I suggest DANISH OIL. It's very easy to work with. I've found that stains are often tricky to apply evenly on a deeply textured surface. Danish oil can be applied directly to a rag and rubbed over the area The beauty of the product is that, after applying, you can go over the surface, once again, with a clean rag to even up any darker areas, giving you beautiful and even results. If you want it darker, you simply apply more coats. It can be mixed to create more exciting colors. I recently refinished a cherry cabinet and used 1 coat dark walnut and 3 coats cherry danish oil. It came out beautifully! Not to mention, Danish oil gives the wood a rich deep look and highlights the grain beautifully. As for finishing, I think it depends on the look you want to achieve. I've always been partial to laquer or tung oil, but they can require multiple coats, using steel wool between, but are simply beautiful(For non-floor surfaces.) When applying tung oil you can apply with a rag, as well. Laquer is best applied by spraying. I aviod brushes completely.

LisaR

Do not paint the oak stairs.Also don't use
danish oil it's to soft a product for your
project.You should use a poly-varnish either
oil base or latex.It's recommended for hard
wood floors.

If you are relly into the look of the white and stained wood combo another way to do it without painting0which chips and pits after wear and tear(you'll have to constantly paint touchups and wash wear dirty) you could use a white pickling stain on the risers-only thing is you will see any imperfections that are normally unseen from painting. Then varnish over it which will protect the wood for the most part from scuffs and wearing. I use latex products-easier clean up-but some aren't as durable as an oil based product-just have to test them out. Usually whatever a floor finisher uses I would use-just add 1-2 more coats, considering these are stairs, and probably would see more wear and tear then certain areas of your floor. GOod luck!