Is there a specific reason why you need a car? It is much more relaxing to hop on the bus and 17 minutes later on the direct train to Bern.Traffic jams on the A1 motorway may occur.
If you need a car: Yes, it is cheaper to hire the car on the french side. Ask for a car equipped with the swiss motorway sticker. From the city of Basel to the city of Berne you almost never need snow chains at any time in winter; If you should like to go to the Bernese mountains, then it might be a different story in the mountains. But even in the mountains: Mostly no chains.
But in any case: make sure you got winter tyres! Before christmas it is even unlikely to have snow on the motorway, but everything below 6 degrees celsius is ideal for winter tyres. (lots of people don't know).