If you're looking for a meeting place or a good stop around ½ or halfway between Basel and Bern, you can try Derendingen, Switzerland, which is almost an hour south of Basel, or about half an hour north of Bern.
Another possible place to stop near the midpoint is Wiedlisbach.
About ¾ or three fourths of the way, you can stop at Urtenen, Switzerland. This is almost an hour south of Basel, or about 18 minutes north of Bern.
If you drive straight through without stopping at all, the total distance is 110 km. The distances shown above don't include local roads to get into each town, so if you're planning a long road trip, make sure you factor in traffic and road conditions, rest stop breaks, and interesting detours along the way. If you're in a rush, you can do the trip with zero stops, and your total travel time would be around 1 hour, 17 minutes.
