I live in California but I still use a stabilizer in my diesel fuel tank. I don't work about fuel breaking down, whatever that means. I do worry about algae growth in the fuel tank. Cheap insurance to add a few ounces of fuel stabilizer and a $14 lasts me all year. Algae does grow inside diesel fuel tanks in California - ask people with diesel engines in their boats that sit for any period of time.
Biodiesel is another concern as this changes the dynamics considerably. Biodiesel will oxidize and this can create problems with clogging of the fuel system. It is recommended to drain the fuel system and tank completely or to put in petroleum diesel fuel before storing a vehicle for any period of time. Problem is knowing if you are getting 100% petroleum diesel at the pump. Better to be safe by adding a stabilizer.