Personally, I never clean my horns, but if you want to, here is what I would do. My assumption here is that the horn is smooth and you are just wanting to remove accumulated grease and grime.
Use a soft piece of cloth. Dampen it and add a bit of liquid dishwashing detergent (I use Dawn in my kitchen, so that is what I would use for this. I really don't think it matters which brand all that much.). Wash the horn until the accumulated dirt, grease and grime is gone. Then with another soft cloth, dry and buff the horn. Set aside and let air dry completely.
Get a can of hardwood floor paste wax (I happen to prefer MinWax, but any good wax will do). Scoop a bit out and rub it onto the horn until all the surfaces are covered. You don't need a lot, just enough to thoroughly cover the horn. I just use my hand for this, but a soft cloth will also do. When the wax is completely dry, come back again with another soft cloth and buff to a nice luster.
Soft cloths are fine, but I have found the blue paper shop towels you can get at hardware or auto supply stores work great.
Should your horn have fine work on it, like engrailing, etc. you will want a soft bristled toothbrush to clean the dry paste wax out of those areas.
Good advice from Ghillie regarding the carpet beetles. You can also use moth balls to protect from them. A little dab will do ya, so don't overdo it. I imagine the no pest strip would smell better.
OK, that's what I would do if I were to clean my horns, which I don't. And don't forget, YMMV.
John (Bigsmoke)