As someone calling BSL home, what I do mostly these days is fly to FRA and then take a three-hour journey to Basel on an ICE high-speed train from there. This has turned out to be the most comfortable way of getting there. FRA has a long-distance ICE train station directly in the airport with direct connections to Basel every two hours, and more if you are willing to change trains in Mannheim (get off, board another train across the platform, very seamless if everything is on time).
Example: UA960 arrives into FRA at 9:15. Direct train leaves at 9:51, you're right in the city center of Basel at 12:47.
Or LH 403 arrives at 7:35. Train leaves at 8:09, and you're in Basel by 10:54.
LH 403 is on a 747-8, and the upper deck on these is a really nice place to be - especially when you travel with a companion or get an empty seat next to you.

Hey, and since this is FT - you even earn miles for these train segments, depending on the program.