The idea of wealth scares Katniss because her entire identity is built around being the protector and provider for her family. If she no longer had to hunt and struggle for survival, she wouldn't know who she was. Her ideas of self-worth are wrapped around her ability to survive and help her family survive. Katniss has never had money and would not know what to do with it.
She also fears being a winner because of the duties a winner must maintain. She would have to tour the districts and the capital. She would be forced to continue smiling and entertaining the people. Katniss fears she would never again get to be herself if she were to become a wealthy games winner.
With respect to her 'old' identity it is clear that Katniss has taken on the role of sole provider for her family since her father died. She hunts and poaches to feed her family (mother -who suffers from depression and younger sister Prim). Without Katniss her family would probably starve. She is for want of a better phrase - the man of the household.
For Katniss winning is literally surviving, the money gained would definitely offer her family a better life, one of security and stability (not living from hand to mouth each day, unsure of the next meal). However, the fame that comes with the fortune is something which I believe Katniss despises, as it represents the capitol. She is extremely uncomfortable at the thought of being paraded as a trophy representing the ruling body which is the establishment. It goes against everything she has been living for.