What's the best way to finish new interior steps.

http://homeguides.sfgate.com/apply-wood-finish-stairs-47190.html

Stain, paint, cover with laminate (they have end caps for steps), tile, stencil, carpet.....find pictures you like on Pinterest, then copy it. Usually though, to make you eye flow throughout the house, stairs are finished similar to whatever the flooring is. ie: if you have carpeting throughout, then the stairs are carpeted.

If you have children, the best way is carpeting. Wood is hard and slippery, and that looks like a steep staircase.

I personally like the look and feel of carpeted steps, but it really depends on the rest of your floors. If you have wood floors and want to leave a wood step, I would stain and use a carpeted runner.

You could stick thick cork tiles on them............

Don't carpet if you don't need to - it's so hard to vacuum! Stain or paint! If you have little kids just teach them to go slow and use handrail. You can always add something on tops to keep from slipping. Just painted my stairs (because I couldn't stain - they weren't nice like yours) and I love them! Just sweep to clean or wipe down, so easy! You could stain the tops dark and paint the backs white - this is a beautiful look. The dark tops hide marks too.

Sand, stain, and varnish