suggestions for 10 nights in northern spain in June 2019 without a car

San Sebastian? Or nearby Hondarribia. Either would make one good base for the area near to France and the French Basque area.Make sure you visit the picturesque villages in the foothills of the Pyrenees, Ainhoa, Sare, Ascain, Espellette, etc. Easily reached from Spain as day trips. I might give this base more than five days.
Choose another town to the west such as Getaria for another base. It is such an interesting area.
Read Michael Kurlansky's "The Basque History of the World" before your leave home as it will help you to understand this ancient culture.

I believe you will find day-trips by public transportation a bit challenging if you base in small towns. The buses and trains tend to fan out from the main cities, so unless your starting and ending points are on the same bus or rail line, you may need to head into San Sebastian, Bilbao or Vitoria-Gasteiz and transfer. That could lead to a lot of transportation time on the way to destinations that only have enough sights to keep you occupied for an hour or two.

I would imagine that public-transportation day-trips from Hondarribia to the French side of the border might also be challenging. I suggest researching the availability of bus and train service before committing to that plan. You can start with Rome2Rio.com and drill down to find links to the appropriate train and bus companies' websites. Don't trust the travel times, frequencies or fares provided by Rome2Rio itself.

I do like Hondarribia a lot. There's a fishermen's district (with restaurants) down along the waterfront as well as the historic center up on the hill. It's not so tiny that I would object to spending a night or two there. Five nights seems like quite a long time, but you've said you are not interested in larger cities.

Zarautz/Getaria/Zumaia are three former fishing villages that can be visited easily from San Sebastian via a combination of train and bus. I even walked between two of them.

The fastest way from San Sebastian to Bilbao (which, though larger, is less touristy and has a bigger historic district) is by bus. The train takes much longer but does go through pretty countryside.

Vitoria-Gasteiz is the capital of the interior Basque province. Not a small town but very pleasant (again with an impressive historic district on a hill) and relatively non-touristy. However, I've only visited it once and had the impression that lodging there isn't cheap. Perhaps I was unlucky in my timing; I didn't try to find a hotel room until 2 or 3 days before my planned arrival.

Although you plan a summer trip, you may encounter iffy weather. The Basque Country tends to be cool and overcast (with and without rain) even in mid-summer. If you are unlucky with the weather, you may have difficulty occupying yourselves if you stay in small towns. I traveled through that area in June 2016 and generally saw few people in the water--and most of those were in wetsuits. I didn't get rained on much, but it was cloudy and cool (a very pleasant change after the sweltering weather farther south).

I hope my information is still useful since my own trip to this area was almost ten years ago. And also late June, early July, so my weather was beautiful. It remains one of my most favorite travel experiences! I found the RS guidebook ( including the tour itinerary) very helpful for planning but as usual, the RS hand drawn maps, not so much! I used the train, regional buses and an incredible one- day private car tour for a hike and seeing the coast, in Spain. Basically everything is a day trip from your chosen base, I suggest one location for the French side and one location for the Spanish.
Travel is so subjective, but I loved San Sebastain for my base in Spain and stayed in St. Jean de Luz for France. From there I then stayed overnight in Bayonne ( incredible cultural museum) and took the train with tons of pilgrims to St Jean Pied de Port. RS doesn't make it sound very appealing in the book, for me it was an absolute highlight. I even walked a bit of the Camino, since the " gate" is there.
If you are walkers or hikers you could even do some of that.
I think I had 10 days.
Language- I can speak French, but that was only helpful on the French side. At the border and in Spain (I can't speak Spanish) navigating was sometimes problematic, but that was pre Google. In any case, Euskera was displayed first.
Learning the history and culture prior to your trip IMHO is a must to at least try to understand this area! Hope this helps!

Logroño? Not even a small city, at best big town. As it is the capital of La Rioja, it has decent connections to Victoria and Pamplona. There is also a good region bus service with allows day trips to some of the villages in La Rioja, like Haro, Santo Domingo, the Dinastia Vivancos Wine Museum and San Millan, birthplace Spanish (and Basque) and is a UNESCO heritage site. And there is a easy access to Laguardia in Alava. Plus in Logroño you have a great tapa offerings in Calle Laurel, Calle San Juan and the "Calle Laurel Barato" each of which would provide a night of tapas and wine (and compared to other places they are relatively cheap). From the 7th to the 12th of June, Logroño celebrates it's patron saint, St. Bernabe. The whole town goes medieval, the streets become full of market stalls and there is a lot of activities. In June 1521 the city was under siege, and on the 11th it was lifted. During the fiesta they have reenactments.
After a few nights in Logroño, going up to Hondarribia would be a good plan.

As for the weather, there is a saying here: Hasta el 40 de mayo no te quites el sayo (Don't take your tunic/cape off until the 40th of May). Basically, the first part of the month can be unpredictable. The rest of the month should be nice.