You can use any number of dyes, to color the antler. When you are finished staining the antler, coat the horn with varnish, or lacquer to protect the color from wearing off. When the varnish or lacquer dried, you can maintain it by giving the measure a coat or two of wax after every use. Any wax works, from furniture wax, to car, and boot waxes, to expensive antique waxes. Use what you already have.
Because Antler is actually made up of minerals, its NOT porous like wood. The Stain is not going to " sink" in deeply on an antler. That is why you need to cover it with a protective finish like varnish or lacquer.
Marine Varnishes have special Ultra Violet light blockers in them, so that the varnish does not "Yellow" over time and exposure to sunlight. If you decide to use a varnish, make sure its for outdoor use, and has the blockers in it.
Lacquers don't have this, and a lacquer finish can become "cloudy" if exposed to sunlight. But, Lacquer can be easily removed with Alcohol, so that you can re-stain if desired, and then re- coat the antler with a new finish.
I have simply used stock finish "oils" to protect antlers I have used to make knife handles in the past. When the " oil " dries, I coat the handles with wax, and then buff the handle. When I am cleaning up my gear after a trip to the field, or range, I put more wax on the handles before storing the knives. ( Oil the blade, after cleaning them off of any organic matter.) The Sheaths get some Neetsfoot Oil treatment if they seem to be drying out. If a leather sheath or other gear is already dried, I take out my saddle soap and give the leather a good rubbing before it goes into my closet for storing.
Take your choice. The leather dye sold by shoe repair shops for dying shoes and boots seems to work on antlers as well as anything else I have tried, or seen used.