Sorry, I must have misunderstood your post. Water is not a good idea for cleaning any bare wood surface. Others here may have better ideas, but I'd start with Denatured Alcohol and move on to Lacquer Thinner. If those don't take it off, your next option is to sand them out. Further, you should take the time to sand the whole thing anyway. I'd use 100 Grit sandpaper for this.
For finishing I tend to stick with oil or alkyd based clear finishes. Stain the wood if you want a darker wood, then finish the treads of the staircase with either a floor varnish like McCloskey's GymSeal or Varano's Black Diamond, or if you want to get fancy you could use Fabulon, or AquaFabulon which is a two part epoxy floor finish(there are many options here, consult a quality paint store, e.g. Sherwin-Williams, or Benjamin Moore). Apply at least 3 coats to be sure you have a durable finish, sanding between each coat with fine sandpaper and cleaning up the dust with a tack rag.
For spindles and railings my favorite finish is Benjamin Moores Low-Lustre Polyurethane. Apply at least 2 coats to the spindles and 3 to the railing. Take the time to sand with fine sandpaper between each coat and clean up the dust with a tack rag. Strain the final coat through a "paper cone" strainer and apply when no one is around to stir up dust. When it's thoroughly dry (at least 2 days), polish the railing with a piece of wet felt and a paste made from ground pumice and water. Rub it with some pressure and elbow grease, then clean up the residue with a soft cloth and clean water. This last step should make it as smooth as glass.