If you ever see yourself but wanting the runner do them all the way across. That is the pro of doing full tags and risers. down side is your carpet sits higher than the risers and treads. if you go with the caps, the carpet and caps will roughly finish at the same height (i prefer this look when a client wants a runner).

Either way works but always be sure to use a nice thick rug pad on the treads as it will extend the life of the stair runner and help protect the wood if you do the full step.

Stair runners can impose more danger on stairs if not fastened securely. So you have to make sure that you do it right.