Sounds like fin rot. Usually caused by Aeromonas or Pseudomonas bacteria, which take advantage of fish in poor water conditions or low temps. What's your ammonia/nitrite/nitrate levels? Whats the tank temps? Thumbs up on the water change; how often do you do water changes? And do you gravel vac each time?
You'll want to keep water conditions pristine, and start using a broad spectrum antibiotic like tetracycline or kanacyn. Keep in mind antibiotics will likely muck up your bacterial colonies, so you'll want to test the water frequently. This is one of the few occasions where I would recommend Amquel or another ammonia converting product as you'll want to keep the medication levels up in the tank and water changes make it difficult to work out how much meds to add to keep those levels even. You'll need a salicyte test to make sure there is no unconverted ammonia in the tank (the Nesslers can't tell the difference between converted and unconverted ammonia). Small amouns of salt are also recommended; to help the fish fight the infection as well as to treat possible nitrite poisoning if the bacterial colonies go belly up. Maybe a teaspoon per 5g, as tetras don't do well with high levels of salt.
