Basque country - recommendations for home base city for day trips?

I don't know anything about the three cities/towns you mentioned, so I can't comment on those.

Is Bayonne too big? Is Saint-Jean-de-Luz too touristy?

I would recommend taking a look at Gernika.

I suggest using your preferred site for identifying/booking accommodation and set the map to cover the area of interest, filter for amenities, price, etc., and make a list of places that suit your needs (maybe with a sea view, free parking, a rate that balances the car rental?). Then take a look at those towns to see if they have what you're looking for in terms of infrastructure.

We found Getaria to be ideal. The coastline is beautiful, we walked to Zarautz and Zumaia. Good bus connection to Donosita. We planned on 3 nites, stayed 4, never moved the car.

Brad

Getaria and Zumaia were both very nice to visit. Close to them is Hondaribbia that was also nice. Getaria and Zumaia were only stops en route to Hondarrbia. We only spent nights in Hondarrbia, at the Parador. As we had a rental car, I can’t speak to which has better public transportation options.

Where do you arrive/depart? You could consider two bases, for east and west (I'd go east without car, west with one, dropping it at Bilbao). The area is somewhat accessible with bus and train, but especially if wanting to explore inland from the coast, you will likely want a car, so I would make your personal wish list first, then determine the bases within the driving distance that you prefer.
We did not have a car but next time would choose to have one. We chose to stay in Deba because we wanted easy access to the coastal trail, but then we wished we had just stayed in San Sebastian longer and used the train to access it at Zumaia.
We stayed near Bermeo in Mundaka and it's a gorgeous area. We only passed through Lekeitio but I have read good things about it. Elanxtobe was magical but just stayed a night--in off season it was just too quiet (and our hotel felt like a haunted house out of Scooby Doo). I think I would choose Getaria on a return visit--good middle ground, known for excellent seafood.

We used Hondarribia as a base and it was great. Easy to get to San Sebastian via bus, and also easy to get to French Basque towns as well. And Hondarribia is a beautiful and interesting place in and of itself, with great restaurants and some very nice hotels.

I've stayed in Getaria but we had a car, which would be helpful there and in the other towns you mentioned.

The port area of Getaria is lovely (but does draw many tourists in season) but there are some beautiful places to stay in the hills just above the port, in San Prudentzio...some are wineries and others not, but Hotel Iturregi is among the loveliest hotels I've ever visited in the Basque region. The accommodation standard in that area is very high, even in the simplest, rural inns.....

https://www.hoteliturregi.com/en/

In Zumaia:

https://jesuskoa.net

Are you set on Biarritz???

Not far from Bilbao but you'd need a car, marvelous rural hotel like your own, cozy house. We used this as a base for a few wonderful places to eat in the immediate area...one of these is: REMENETXE. Lovely walks in the green hillsides, just a gem......

http://www.urunehotela.com/en/slow-life-3/who-are-we/

Bilbao for at least two or three nights, so you can use public transportation to visit Getxo (the old fishing port and Puente Bizkaia) and Plentzia (both on our great, clean, safest, cheap subway system), and to visit Bermeo, Mundaka and Gernika by bus or on the Euskotren train. Then take the bus to Donostia-San Sebastian, and then easy day trips on the public bus (again, clean, safest, cheapest, frequent) to Getaria and Hondarribia. Biarritz has a famous name, but nothing much there, IMO. Baiona-Bayonne is much more interesting.

We travelled the Basque country by car and stayed in Bilbao. San Sebastián, Hondaribbia and Bayonne. Also had stops in Gernika, Getaria, the French Basque towns, St. Jean de Luz, and Biarritz. With just 6 days, would suggest 3 days in/around Bilbao and 3 days in/around Hondaribbia (and would skip Biarritz). You will likely want to return for more!

We did the same. We split between Bilbao and Hondarribia and it was perfect.

Thank you to everyone who responded so promptly and with such helpful information! You've made planning for our trip far easier, and we know we'll benefit from your knowledge as we enjoy our time in Basque country.

Suggestion for a day trip or stop en route from Bilbao to Hondaribbia: https://www.museochillidaleku.com/en/visit-us/practical-information/

Have a great trip!