My wife and I are planning to be in the Basque region from 11/11 to 11/15. We will rent apartments during that time. Would it be better to stay in Bilboa for 2 nights and then in San Sebastian for 2 nights or just stay in San Sebastian for 4 nights and take day trips to Bilboa and other locations?
Two and two.
We spent a week in San Sebastián a few years ago and enjoyed it. It is quite easy to day trip to Bilbao, which we did. Our day trip in Bilbao was enjoyable. Either city would make a fine base. Changing apartments after just 2 nights seems nonsensical to me, especially since you are moving just a short distance. We tend to prefer having 4 or more nights in a city/town of our choosing, especially if there are some good day trip options to mix things up. By the end of your stay, when you get back to your rental after a day of sight seeing, you have just a tiny feeling that you are returning home. It’s a great feeling.
It’s very easy to get between Bilbao and San Sebastián on the bus. When we visited the region we spent four nights in Bilbao with a day trip to San Sebastián and another to Haro in La Rioja for wine tasting.
It´s November, so weather will typically be rainy and cold (although you never know, our weather is very unreliable, but Donostia-San Sebastian gets 65 inches of rain per year, Bilbao 55). Not that much to do or see in Donostia-San Sebastian, beach weather will not be a thing (almost for sure) and I would, instead, recommend to stay in Bilbao and visit D-SS.
In case you decide to stay in Donostia, a day trip to Getaria and another one to Hondarribia is a must. If in Bilbao, Getxo, Plentzia, Bermeo, Gernika...are nice places to visit.
Both cities are very well linked by the bus (recommended) and a commuter train (less practical).
I would do 2 and 2. Bilbao is worth two days - it is just a fantastic town that doesn’t feel overrun with tourists. The old town is a great place to stay (I stay at an airbnb there and it is fantastic).
Amending my reply. Or do 4 in Bilbao and take a bus up to San Sebastián.