Total knee replacement (TKR) often gives patients good pain relief. With less pain they can walk better and farther. Function improves and they can start doing things they haven't done in years (like stair climbing or even dancing).
But studies show there are other problems that aren't solved with a joint replacement. In fact, the surgery may make things worse. For example, loss of quadriceps muscle strength (the large muscle along the front of the knee) makes balance more difficult.
Patients who have both knees replaced are more likely to trip and fall. This is especially true if and when they have to get around objects or obstacles. A program of balance training and exercises to improve joint position sense may be to good place to start.
But first, make sure there isn't a medical reason for your father's loss of balance. Have him see his doctor for a check-up. If there are no medical problems, then give rehab a try.