I would take scraps of the fabric, wet them, put them on a paper towel with the red on top and put a glass on top. Go back in an hour or so and see if there is color transfer. The weight helps simulate the fabrics held together by stitching. Since you have darks, you probably will wash in cold, but I would use water just a little warm.
When washing, I would use 1 cup of vinegar instead of one of the synthetics -- and for color catchers I use a couple of white hand towels.