I agree with Gundersen that Interlaken is a good choice for what you want to do: sleep somewhere central with easy access to trains, boats, mountains, etc. and spend your days exploring.
Interlaken, like most tourist locations, consists of a main street with all the shops and touristy things. This is where the masses hang out - go one street to either side of this, or walk along the wanderweg path along the Aare river and you have peaceful Swiss beauty.
If you choose to stay here, I would even suggest Unterseen as a good location. Directly beside Interlaken with no discernible division from it, a lovely little town square and a bit more peace and quiet. We have often rented a bnb apartment there in the past.
Between Basel and Bern I would for sure choose Bern, as Basel is at the edge of the country and Bern more central.
Travel times from Bern to Zurich, Lucerne and Lausanne are all around one hour. You can reach the mountain village of Wengen in an hour and a half.
You can also check out times and distances from Thun. I find it very fast and convenient to take day trips from here in all directions.
Use SBB.ch to explore your train, bus and boat options.