I would buy a small can of birch stain (which won't be much darker than the pine) and apply a small amount to the pine. It should be applied with a cloth to control the amount. Start with a very light amount and then wipe it off to achieve the desired shade. This way you can compare it to the existing birch. Of course, use only a small scrap of the pine until you get the right shade and be sure to let it dry thoroughly since the shade may shift to darker as it dries. Use a good quality stain. Some stains incorporate a lacquer finish which can dry to rather a high gloss. If that's your desire, fine, but you may only need a low lustre finish... Best of luck!