First, it is important to define static versus dynamic characters. A static character does not change over the course of a story and remains essentially the same at the end. A dynamic character, however, does undergo an important change.
Second, it is vital that you make your own decision about this question, particularly as it could be argued from either standpoint. On one hand, Katniss is a dynamic character, specifically when considering the main conflict of the text--the Districts against the Capitol. Initially, Katniss seems to hate the Capitol but believes that nothing can be done to stop them; while she often tries to rise above this belief during the course of the story, there are several events that quickly stomp out her resolve and, thanks to her thoughts, we are able to see that she frequently considers the ending of Capitol rule an impossibility. However, despite her own doubts, the horrors forced upon her provoke her into action, rather than inaction. At the start of the trilogy, she is very much convinced that the wisest decision would be to run into the woods with Gale and their families and disappear, living off the land. This idea is developed over the course of the trilogy, but in the end, she is determined to bring down the Capitol instead. Therefore, she does experience a change in beliefs and acts upon this change.
On the other hand, you could argue that her rebellious thoughts and nature, evident from start to finish (for example, the trick with the berries, shooting the arrow at the Gamemakers, even hunting illegally), indicate that she had no choice but to continue in her way and lead the uprisings. This would technically make her a static character, considering she is consistently rebellious throughout the trilogy.