The easiest way to stain lighter-colored sapwood so it blends with the heartwood is to stain all the wood with a dye stain, as is shown on the right side of the accompanying picture of walnut. Dye is much more effective than pigment, or any commercial stain that contains a binder.
You can apply the dye several times after each application has dried to get a darker color. Or, with water-soluble powder dyes, such as those from W.D. Lockwood, you can wipe over the stain with a water-damp cloth to remove some of the dye and lighten the color.
You can also apply more of the dye just to the sapwood, after you have stained all the wood, to selectively darken it and achieve a closer blend. But applying dye just to sapwood, and not to all the wood, seldom works well because it’s so hard to get a match – though you could always try it. The worst that would happen is that you would then apply the dye to the entire surface.