Cindy,
As far as where you should fly out of the US, the NYC airports (either JFK or Newark/EWR) are generally the cheapest for flights to Europe. Conveniently, transatlantic flights almost exclusively leave after 9pm, so you can book a cheap morning flight into NYC from your home city and have plenty of time to make the connection.
As far as destination airports, Donostia/San Sebastian, Biarritz, and Bilbao are the closest to the start— you can take a <2 hour bus or train to Irún/Hendaye from any of those. Bordeaux is ~4 hours away from Hendaye by train. None of those accept direct flights from the US, so you’ll have a layover somewhere if you choose to fly to one of those.
Places with direct flights from the US to consider flying into would basically be Paris (~6 hour train to Hendaye) and Madrid (~5.5 hour bus to Irún).
My partner and I are flying from Minneapolis to JFK (NYC) in the morning, then JFK to Bilbao (overnight flight with a layover in Lisbon), for example. We will arrive at Bilbao in the afternoon, take the bus from BIO airport to San Sebastián (~1 hour 15 min) and stay overnight, then will take the train (40 min) to the starting point in Irún the next morning.
Use Google Flights to search for flights. Flight Connections will show you which airports you can fly to directly. Rome2Rio searches all kinds of transportation to find the best route between two places. Hope that helps!