Sounds a heck of a lot like fin rot. Use Jungle Anti-Fungus (aka JFE) it treats fin rot and fungus. Melafix will NOT (IMO) stop the disease and the rot may spread to his body. Treat him ASAP.
You can keep up the Melafix at the same time, it will help with re-growth of the fin. It will take a few weeks for his tail to grow back.
Fin-rot is not red, it looks whiteish, like dead skin. If you betta's fins are already white in color or opaque, you may not see it. Most fin rot is fungal but there is also a bacterial form of rot as well.
I have had good sucess when treating with Jungle Anti-Fungus - they are little yellow crystals you add to the tank, makes the water gold in color, and it should do the trick. Don't delay, fin-rot is not as deadly as body-rot.