If your finch looks depressed, has odd colored diarrhea (yellow or bright green), and discharge from the eyes or mouth, it probably has a bacterial infection. If untreated the finch could die. The color of the droppings and discharge from the eyes or mouth are probably the best indication of a bacterial infection. The only way to determine exact which bacteria are infecting your finch is to have a vet run a few cultures.
Now I should point out that not all bacteria are bad. Keeping your finch's environment sterile may only do more damage in the long run. All living animals (birds and humans too), have various types of bacteria in their bodies and on their skin. These bacteria help to protect the bird from infections and to make digestion of food possible. A finch with no bacteria in its system would die very quickly.
Most all finches have a few of the potentially harmful bacteria in their systems at any given time. If they have a good diet and a healthy home their immune systems can handle these bacteria. In times of stress however, the immune system may not be able to handle the bacteria and this when an illness develops. Moving, breeding, or other illnesses can trigger the infection to spread.
All bacterial infections are contagious. Always isolate an infected finch as soon as possible. If the finch lives in an aviary setting and you worry you didn't catch the symptoms in time; you may want to treat the entire flock. During the breeding season is when a bacterial infection can do the most damage. Parent finches transfer both good and bad bacteria to their young which are born sterile. If the chicks develop a severe infection before their immune systems are ready to handle it they will die. Chicks that dies for no apparent reason before it leaves the nest or just after fledging is often the victim of a bacterial infection.
For a treatment, you can consult your vet. Keep in mind the vet will need to see the finch and will probably need to run a culture. This can be expensive and is usually quite stressful for the sick finch. While I think vets are great I have often found it easier to just treat the illness myself. In most cases if the finch isn't treated within 48 hours of the onset of symptoms, it will die.
The best treatment available today is the antibiotic Amoxitex or Tetratex. They are very powerful medications and should be given for 5 days in a row. No less, no more. Make sure the medicated water is the only source of water in the cage.
Amoxitex may also be given directly into the crop. Mix 3 grams in to 10ml (or 2 teaspoons) of water. Dose at a rate of 1ml for every 100g of your bird's body weight. Now, keep in mind that finches don't weigh 100 grams. In fact most zebras average around 12 grams. For a finch the amount given via the crop would be only a small fraction of that amount. I'll put this into CC because we use 1 CC syringes to dose crop medications. It's about up to the 4 on a 1cc syringe. This is given once a day for 5 days.
Because Amoxitex is such powerful stuff it tends to kill the good bacteria along with the bad. After using Amoxitex for the full 5-day treatment, you should always follow up with Probotic. Never give Probotic while you are administering any antibiotic.
Probotic is a live culture of healthy bacteria. It can be used on cats, dogs, cage birds, and poultry chicks. The dosage for birds is 1gram (the scoop which comes with this product is equal to 4 grams) per Liter of water. Give as the only source of water in the cage. The mixture should be given for a few days (2-3) and that is all. Probotic can also be used to treat most any type of gastrointestinal distress.