Laura, Use Basel or Zürich, whichever gives you better times and /or prices. As you say, the time from either of them to Interlaken is similar and less than from Geneva. Basel airport has a shuttle bus to Basel station, takes ~15 minutes, not a problem, you can buy tickets from Basel airport to Interlaken (or any other Swiss station) which include bus + train. SBB ticket machine at the bus stop.
If you have time, take the route via Luzern and the Brünig pass line instead of via Bern. You can do this from Basel or Zürich. The fastest route is via Bern, and that is what you get by default. The Luzern+Brünig route is scenic, three lakes and a pass. This route is not possible from Geneva.