Quick selection tool leaves red line around selected area when done

A couple of answers depending on your subject.

1. For smooth hard edges I would make a path with the pen tool and convert the path to a selection. It will give a better selection on that edge than the quick select tool.

2. With any selection, but particularly those with a softer /anti-aliased edge, the edge pixels can be contaminated with colour from the previous background. This is easily resolved. Add a layer above the masked layer and set its blending mode to colour. Then Alt click between the layers in the layer panel so that the new layer is clipped to the layer below. The clipping means it will only affect pixels that are solid on the layer below.
Now take a small soft edge brush and pick up the colour from the edge of the masked image by Alt clicking (you may need to change this as you work around the image) then brush around the edge of the selection. The red contamination will change to the image colour.

Dave