Slowly and methodically.
For starters:
Create legislation that prevents construction firms from making campaign contribution and MTA employees from taking jobs as political consultants. This is probably the largest source of corruption, graft and waste.[1]
The MTA's Board of Directors is predominantly controlled by the Governor of the state. The Board of Directors has long been a political dumping ground. Residents - the people who depend on the train have almost no say. Residents should have representation on the board of directors even if its a only a handful of rotating seats. Having people on the board who actually use and depend on the train would provide an instant feedback loop.
I think you also need to implement a rotating independent oversight committee whom the MTA must be beholden too. One that is democratically elected. The oversight committee should have real teeth though perhaps with veto ability in certain instances.
And yes as someone else mentioned you probably need to fire a whole bunch of people.
[1] https://www.nytimes.com/2017/12/28/nyregion/new-york-subway-...