Swiss Railways (SBB/CFF/FFS) operates a train from Bern to Basel hourly. Tickets cost $35 - $70 and the journey takes 55 min. Alternatively, FlixBus operates a bus from Bern to Basel twice daily. Tickets cost $5 - $10 and the journey takes 1 h 10 min.
From Bern train to Basel SBB. 55 min SFr 63. From Bern bus to Basel, Basel central station. 1 h 10 min SFr 8. From Bern drive to Basel. 1 h 2 min SFr 14. From Berne rideshare to Basel. 1 h 23 min SFr 5.