If you worry a plant could be a culprit, take out the plants, and check them all. Any rougher plants, could damage his fins. Stick to "fabric" or silk plants, and gentler plastic plants :)
Also it doesn't look like fin rot "to me"... Go get him a heater, for one. This will keep away stuff like fin rot, popeye, and all that, along with proper water changes. You can do 50% twice a week for that tank size.
If his fins keep getting worse, with all sharp or rouogh plants out, and the heater on, then you can get AQ salt, which'll help a lot! stay away from bettafix and mellafix, as it is basically poisonous to bettas.