Border-crossing ground transportation is often not terribly frequent, and it may require transfers. Every transfer tends to cost time, which you end up spending inside the bus or train station. However, Rome2Rio.com indicates that there are multiple bus companies making the run from Bordeaux to San Sebastian and/or Bilbao, in addition to trains. Without digging down into the actual schedules, the Spanish Basque country looks do-able to me from Bordeaux.
I haven't seen the French Basque Country yet, but Rick covers it along with the territory on the Spanish side of the border, so you can check out his suggestions. I believe he had positive things to say about St. Jean-de-Luz, and I think someone posted here not too many months ago that St. Jean-de-Luz was a favorite of his/her tour group.
In Spain I enjoyed both Bilbao and San Sebastian. I made side-trips via bus and FEVE train to Hondarribia (very attractive, nearly on the French border), Zarautz-Getaria-Zumaia, and Bermeo--the latter from Bilbao, the others from San Sebastian. The Spanish Basque countryside is very nice, too.