I think either will work, and the timing is good since those regions tend to be much cooler than the rest of Spain during the summer. A week is a reasonable amount of time, but be careful about transportation in and out, as that could drain a fair amount of important sightseeing time.
I traveled to both areas in 2016, using public transportation. Both have lovely countryside, but I'd give a slight edge to the Basque Country in terms of inland scenery; the best of what I saw was a bit more dramatic. Galicia probably wins for coastal scenery. If pushed, I'd say I slightly preferred the coastal towns I visited in Galicia, but I only saw a few in each area, and there's no shortage of charm in any case. I have no interest in beaches, but even I noticed San Sebastian's beach.
I know nothing about wine. Seafood--especially shellfish--is, if anything, even bigger in Galicia than the Basque Country, I think. If you happen to be foodies who don't much care for shellfish, the Basque Country might be the direction you should tilt.
A few comments on the places I visited in each area:
Basque Country
San Sebastian is very touristy, Bilbao somewhat less so, the inland capital of Vitoria-Gasteiz even less visited by foreign tourists. Bilbao and V-G have more museums but SS has at least one good one. SS has the smallest historic district, and it's pretty weighted toward tapas bars and restaurants packed solid with (mostly, it seemed) tourists. V-G has a large hilltop historic district, charmingly called "the medieval almond" because of its shape. It has an historic cathedral that was under restoration in 2016 and offered English-language tours.
Among the smaller towns in the Basque Country I thought Laguardia was the most atmospheric. It's a hilltown near the LaRioja wine country. I noticed several wine bars as I wandered though the town. You can get to Laguardia from V-G by bus, because it's on the line to Logrono, but the buses are not terribly frequent. The church is supposed to be interesting, but I think you have to get the key from the tourist office, and I happened to show up on a day when it was closed due to a local fiesta.
The relatively large town of Hondarribia, over near the French border (and the SS airport), has a fishermen's district along the water (restaurant central) and an older historic district uphill. It takes awhile to see the areas of interest. You get there from SS by bus. There's a trio of small coastal villages (now developed for tourism) west of San Sebastian: Zarautz, Getaria and Zumaia. You can see them all in one day by using buses and trains. I walked one leg of the trip. Over about midway between SS and Bilbao is Lekeitio, often recommended by guidebooks; I didn't get there. Closer to Bilbao is Bermeo, which has a colorful waterfront. It's accessible from Bilbao by either bus or train.
The train from Bilbao to SS is very slow but traverses some very pretty country. The bus is faster. I preferred Bilbao by a considerable margin, but extroverts/bar-hoppers would love SS, though I don't know how many native Basques they'd meet.
Galicia
Santiago de Compostela is a gem: really large historic district in addition to the cathedral. There are multiple museums as well; it would benefit from 1-1/2 days, I'd think. The place is awash in pilgrims, so not necessarily the best place to get in touch with Galicia. Pontevedra and A Coruna are other largish places of interest; both can get hot. Those are all good bases. Skip Vigo unless you need it for transportation reasons.
Betanzos was my favorite of the somewhat smaller spots I visited; it's worth at least half a day, just for walking around. The small coastal towns of Combarra and Cambados are picturesque but quicker to visit. The inlets (rias) along the western coast are pretty. I liked the small inland border town of Tui; the cathedral has an English audio guide (typical in Spain).
