How to cover old stair risers?
If those refinished stair treads are a common type type of hardwood such as oak, maple or beech you can get 1/4 inch thick plywood with an hardwood veneer on one or both sides. Looks like one sheet of the material would be more than enough to to make your riser skins. 

Precut the pieces to fit without fastening right away. When you cut them you will want the grain direction of the veneer to be horizontal for best look. Mark each piece on the back as to which step it corresponds to and the which side is up. 

Also cut yourself several small pieces of the plywood to be used for color and finish samples. Experiment with these pieces till you get an idea of the finish that looks correct against the re-finished treads. 

Once you like your finish samples you would be ready to pre-finish the riser skim pieces on the good veneer face side. Doing so works well because you can work in a flat place where it is much easier to do a nice job.

When all the pieces are dry it is time to install the riser skin pieces. If I was doing it I would use a light coating of construction adhesive to glue them in place. You will want to select an adhesive that is compatible with the left over residue from the carpet. Even so you will want to remove as much of that old adhesive as possible. A tool like shown below should work well to scape it off.



Take note that the blade will dull so be prepared with spare ones or cleaning up and re-sharpening them. With a good scraper and a bit of elbow grease you can most likely remove all the old adhesive down to bare wood. This would simplify the selection of construction adhesive for gluing in the riser skins.   

It may take some clever fixturing or clamping to hold the risers in place for a few hours while the adhesive sets up. Alternatively you could use some colored hard steel "paneling nails" to tack them in place. With careful selection of where you place the nails they may very well not be distracting. Using a nail gun may mar the surface of the riser skin and make a more noticeable scar than using the type of nails that I suggested. 

It is my belief that if you fit the riser skins carefully there should be no need to add moldings along either the top edge or the bottom edge of the riser skin. You can favor the fit to lower edge skin even using a small plane to get a perfect fit. Then tip the piece into position under the nose of the next tread above. Even if you have a 1/16 inch gap along the top edge no one will be looking intensely under there anyway.