Our family really enjoyed staying in Saint-Jean-de-Luz in French Basque country last summer. It had a family feel, and it was not crowded and touristy. We especially enjoyed the market and walking around and seeing the sunset at the beach. The water was warm in June. We could have daytripped to Bayonne or Biarritz, but we were very content to just stay there.
After that, we stayed in San Sebastian in an apartment for five days. We stayed in the neighborhood near the Zurriola beach, which was less crowded than the LaConcha area and the old town...but easy to walk around to those areas. We enjoyed watching the surfers and saw beautiful sunsets there. Our (young adult) kids really enjoyed the pinxtos experience-we all did!
We did a cider house and cheese farm tour that was memorable and really delicious. The short bus ride took us right up into the mountains and we felt that we were in Switzerland! On another day we took the funicular up to Monte Igueldo for spectacular views. We also spent a few euro and took a wooden boat ride on the Rio Misterioso up there as part of it. That was fun!
We had a little more than a week in the French Basque and Spanish Basque regions and would’ve happily stayed longer and explored more. We took a train to Barcelona and flew home from there. Barcelona would be a great city to explore with your family as well!
Before we got to the Basque country, we visited friends in the Dordogne and Lot regions of France. That was a beautiful part of the country, and I would recommend visiting Cahors, Saint-Cirq-Lapopie, La Roche Gagnac, and Domme. There is so much to see there!
I’d love to return to that northern Spain region again and explore the area between San Sebastián and Bilbao, and take some day trips out of Barcelona. Never enough time!
Good luck with your planning!