I can't help you much as I'm not an expert, but bettas should be in the temperature range 78-79. You should get a heater if you don't have one. If it is fin rot his fins and tail will be black or red on the edges and I think it usually goes from the outside of the tail towards the body. It might be tail biting if not fin rot. The experts can help you more, but if you post some pics and fill out this form, people can help you more. Good Luck.
What size is your tank?
What temperature is your tank?
Does your tank have a filter?
Does your tank have an air stone or other type of aeration?
Is your tank heated?
What tank mates does your betta fish live with?
What type of food do you feed your betta fish?
How often do you feed your betta fish?
How often do you perform a water change?
What percentage of the water do you change when you perform a water change?
What type of additives do you add to the water when you perform a water change?
Have you tested your water? If so, what are the following parameters?
Symptoms and Treatment
How has your betta fish's appearance changed?
How has your betta fish's behavior changed?
When did you start noticing the symptoms?
Have you started treating your fish? If so, how?
Does your fish have any history of being ill?
How old is your fish (approximately)?