Louisville International Airport (SDF), Indianapolis International Airport (IND)
Your two main options for the Kentucky Derby are the Louisville airport (closer) or the Indianapolis airport. Louisville is far closer, but flights can get extremely expensive for the Derby. Many people fly into Indy and rent a car as it is only a two hour drive or so away. If you fly into Louisville you will only be a few miles away from the Derby, so you can simply take an Uber or a cab.
If you rent a car from Indianapolis you are looking at $50 or so a day. An Uber from Louisville airport will be $10 to $15.
Generally the cheapest time to book flights for Kentucky Derby will be about six weeks before Kentucky Derby. This means you should be focused on booking your flight in March 24, 2018. This may seem late to you planners out there, but statistically it's the cheapest time to book flights. That said, if this is too late for you book it earlier. One final note is that plane tickets for Kentucky Derby tend to skyrocket about two weeks out, so certainly don't wait that long to book a Kentucky Derby flight.