Yes, there are many clear finishes that won't change the color. However, you may notice a slightly different tone because sealed wood always looks different due to change in reflectivity...but nothing that should concern you. I like Minwax Wipe-on Polyurethane, which can be easily applied with a rag, thus avoiding the problem of visible brush strokes. I'd choose a satin finish, which will just make it look polished. If you don't seal it, it will start to pick up grime from hands after a while.
There are two issues with what you want to do:
1) Since the stairs are now carpeted, it's going to be dicey applying a finish without getting some on the carpet.
2) Wherever there is wood filler applied to the wood you may have a difference in tone after putting on a sealer. For example, it looks like some fairly large bolt holes were filled on the vertical part of the railing where it meets the floor.
Q