To achieve a narrow binding, don't try to cut or sew the binding narrow. Cut the bias strip wide, (for the one shown here I started with a strip about 2+" wide). Press to take the stretch out, then re-trim. This is a basic trick I learned from my couture pattern making teacher who trained in French couture and it works. The bias strip will stretch a bit and the width will vary. No problem as the next step is to re-trim it to an even width, about 1 1/2-2", making a test sample will give you the best width for your fabric. If you have never done this before, make a few test samples to get the hang of it.
Stitch the bias to the neck edge using a 1/2" seam, pulling with a slight tension so the finished band will hug the neck. Apply more tension about 4" at center front and back. Press flat as sewn, then carefully trim the seam allowance to an even 1/4-3/8" width. Working at the ironing board over a ham, wrap the binding and give a light press. Stitch the shoulder seam closed. Adjust the binding width and place pins in the ditch of the seam. Stitch. Trim the seam on the back side close to the stitching line. While you can turn this edge under in the pressing stage, I left the edge raw so it would be as flat as possible as shown in the photo below.
