Using a damp cloth is the old reliable when it comes to leather care. Since leather is so very naturally durable anyway, giving it a wipe down (without soap — its chemicals can ruin the leather over time) after each use is recommended.
Leather conditioner or cream moisturizes the material so that it doesn’t dry out and crack. This won’t make your Harness “shine,” but it will protect it. Note that in most cases conditioners don’t waterproof your leather (though some do include a water-repelling component). Leather conditioners are lotion-like, and are to be gently rubbed into the leather.
Just like skin, leather needs some ventilation to prevent mildew and rot. Air can naturally pass through leather, leaving moisture to evaporate naturally. Store in a breathable fabric.
Keep leather away from direct sunlight/heat.