More time in San Sebastian
Without a doubt, you must spend more time in San Sebastian and less in Bilbao. Last year, I made the mistake of staying for 4 days in Bilbao and only for 2 days in San Sebastian. It should have been the other way - 4 days in San Sebastian and 2 days in Bilbao.
Bilbao is an industrial city with only the Guggenheim Museum as the main attraction. Having been to other museums in Spain, I can confidently say that art collection at Guggenheim is far inferior than what you can see in other museums.
Yes, the Guggenheim museum building designed by Frank Gehry is awesome, but you can spend only so much time admiring the building.
At best, you can half a day at the museum and the other half exploring the old town area of Casco Viejo, which is full of beautiful Basque architecture and interesting sights.I think you can see all of Bilbao in a day as long as it is not a Monday (When most of the museums in Bilbao are closed)
On the other hand, San Sebastian is far more relaxing and pretty. It has lot to offer for tourists starting with the best of pintxos bars. After my rounds of pintxo bars at Bilbao, I had presumed pintxos were just another tourist hype. Luckily, the Bar Ganbara in San Sebastian corrected my perception. Try gilda and warm croissant with jamón iberico at Bar Ganbara to understand why there is so much hype around pintxos.
The best cider houses are found around San Sebastian and not around Bilbao. Astigarraga, Hernani, Urnieta, and Usúrbil are the cider-producing regions of Basque country and are just minutes away from San Sebastian.
Haro, the capital of Rioja wine country is just 1.5 hours by car from San Sebastian. A tour of Rioja wine country is a perfect day trip from San Sebastian.
San Sebastian has the best of the beaches in the region: La Zurriola Beach for those into surfing and La Concha beach for those looking forward to long walks.
San Sebastian is the hands down winner, if your question is whether to base in Bilbao or San Sebastian.
In five days to visit Basque country, you can make your base (all 5 days) as Bilbao. You can easily visit the entire region from Bilbao because the distances around the Basque country are short.
Another good option is to spend 2 days and 1 night in San Sebastian. After a day’s sightseeing in the city, spend the night and see the French Basque region across the border on the next day. And spend the rest 4 nights in Bilbao.
Bilbao to San Sebastian: 1 hour by bus
From Bilbao to San Sebastian is a 1 hour trip by bus, leaving from the Bilbao bus terminus and arriving at San Sebastian terminus. The road trip is very comfortable on the motorway. You can also easily drive. Do not take the train, it is slow and old.