Hello,
I have a John Deere STX38 that has had a new battery hooked up backward.
No with a new fresh battery hooked up correctly, does nothing when trying to start.
What can be damaged by battery hooked up incorrectly?
Solenoid? Any fusible links on this model? Charging system?
Any help will be greatly appreciated.

Couldn't find a diagram. Any Electronics Modules may be damaged, could be as simple as a fuse or diode.
Ignition switch should be good, as well as starter. Would check fuses, power to ignition switch, then power to starter when switched to Start. With Start circuit going, You want to get Ignition/Fuel Items Sorted, check for spark and fuel.
Now that engine is running, compare running voltage to standing voltage to see if it is charging. I would get the Service Manual, will save Time and Money