Avoid direct sunlight on the skin. Try to keep the humidity and temperature stable.
I store my instrument lying down in hard cases. The bodies are in washi-paper bags. I will often wrap the neck in a saofuki if I am traveling for extra protection, but that’s not necessary. On the other hand, I will put the whole shamisen in a large fabric bag called a nagabukuro for extra safety if it’s looking like rain.
Do not store your instrument with the koma in place. Be careful when you remove it so as not to damage the bachigawa.
You can store them straight up if you like. Traditional cupboards and stands do so. If you store them at an angle, make sure the tenjin is resting on something to keep the weight evenly distributed.
You can lower tension on the strings if you like, but it’s not very necessary.