NB: this tip is valid for "classic" BAdI, not for more recent "Kernel" BAdI. It's also presented in SAP note 1586278 - How to detect BADIs on program execution.
1. Go to the TCode SE24 and enter CL_EXITHANDLER as object type.
2. In 'Display' mode, go to 'Methods' tab.
3. Double click the method 'Get Instance' to display it source code.
4. Set a breakpoint on 'CALL METHOD cl_exithandler=>get_class_name_by_interface'.
5. Then run your transaction.
6. The screen will stop at this method.
7. Check the value of parameter 'EXIT_NAME'. It will show you the BADI for that transaction.