San Francisco to Basel, airlines and connections

I would fly direct from San Francisco to Zurich. You can take the train to Basel from there or take another connection, but you are already at Swiss on first stop, much easier. I don't know when you will travel, but I just checked round trip prices and they are less than 700 USD round trip San Francisco to Zurich in October for economy and likely not that much more for Premium Economy.

The only transatlantic flight to Basel airport is Air Transat from Montréal–Trudeau, and that is seasonal. Otherwise it is Europe, North Africa or the middle East only.
See list here: https://en.wikipedia.org/wiki/EuroAirport_Basel_Mulhouse_Freiburg#Passenger

I agree with LacLeman, if you can get a flight direct to Zürich take that. It is only 1h20 from Zürich airport to Basel by train, direct from the airport station. This is what I do.
Otherwise fly to Frankfurt and take a connecting flight to Basel if there is one at a convenient time, otherwise the train from Frankfurt Airport station to Basel.
I would not change and London Heathrow or Paris Charles de Gaullle. Both airports are too big.

Another option is change at Amsterdam Schiphol. That is a good airport for connections.

don't bother looking for connecting flights.

Option 1 : fly to Zurich, book with Swiss - but book the flight to Basel. This gets the train ticket built into the flight ticket. Direct fast trains from Zurich city centre to Basel city centre every hour with journey of about 80 minutes. Trains are turn up and go, no need to reserve. Make sure the booking is routed via Zurich!

https://www.swiss.com/corporate/EN/company/partners-alliances/airtrain-basel-zurich

Option 2: fly to Frankfurt, book with Lufthansa. Add Rail&Fly to the booking. This gets you a train ticket at a flate rate price which is very good value for the trip to Basel. Direct ICE high speed trains from the airport to Basel city every 2 hours, journey just under 3 hours. In the gaps are alternatives with one change of train. Reservations optional but only really needed Friday afternoons and around holidays (assuming you aren't leading a group!). Basel is considered to be the border station and limit of the ticket.

https://www.lufthansa.com/us/en/Lufthansa-Express-Rail-Fly

other international airlines flying to Germany also offer Rail&Fly, it's just a matter of digging it out of their websites. United codeshare with Lufthansa. But only use Frankfurt flights, the other airports are too far away.

Traveling SFO to Switzerland this past winter we got a pretty good fare from SFO to Zurich non-stop on Swiss Air/United code-share. Train from ZUR Airport with one connection to Basel takes about 1 hr 9 min.

Since you want to fly in premium economy, I would pick the Lufthansa flight that connects in Frankfurt.

Many thanks for the insights and advice. Some years back, before walking was a problem, we used the train often. Now it seems daunting. I had looked at flying into Zurich, and still may consider doing it.

I did not mention that we are looking for an open-jaw itinerary: SFO to Basel; Amsterdam to SFO. Right now, it looks like Lufthansa flights look best, including the aircraft on the long flights, departure and arrival times. I'm a little concerned about connection times in Frankfurt: 1'55" outbound, and 1'20" on the return. It looks like arrival and departure will be from the same terminal. If it's a long walk, and we start from a seat towards the rear, it will be nerve wracking.

There is a longer connection in Munich on the return, but the plane is an Airbus 340, and the reviews of the Premium Economy on that plane are lousy.

So, still weighing the options. Thanks, again, to all.

Check flights on Kayak. If you need an open jaw click on MULTI CITY, then enter:
Date going: SFO > BSL
Date returning: AMS > SFO

Besides the Swissair non stop option SFO-ZRH there are many other one stop options. Numerous flights exist from SFO to major European hubs, from which you can connect to BSL.
Lufthansa and United are partners in the Star alliance and together they have 4 daily flights to FRA and one to MUC. From both you can probably connect to BSL with Lufthansa or SwissAir (SwissAir is part of the Lufthansa Group, and also a Star Alliance nember).
MUC is voted the best airport in Europe, and it is my preferred airport to connect when I fly to Florence from SFO. I don't mind the Lufthansa A340 serving SFO to MUC at all. Actually I prefer it. All their bathrooms (several of them) are together downstairs (they built it in a portion of the cargo hold) so there is rarely a line. You need to go through a short flight of stairs to go to those bathrooms, therefore there is also one large bathroom for disabled persons on the same level as the passengers cabin. Also, the A340 is slightly smaller than the B347 or the A380, therefore the seats on the sides of the two corridors are two, instead of 3, therefore there is no chance of ending up in a middle seat. It's either window or aisle, unless of course you end up in the center seats between the two aisles.
If you connect at FRA, I prefer the Lufthansa flight to the United ones (United makes you pay for alcohol, which is a non starter for me). The LH flight to FRA is aboard an A380. Connecting at FRA is easier now. A few years ago it was a very long walk from intercontinental to the European flights. Now they use the same terminal and the same gates (if your flight is all with Lufthansa), but they use different levels. Upstairs (gates Z) are for intercontinental flights, downstairs are European. So when you deplane the jetway will force you to exit at the Z level upstairs, then you go downstairs (you will go through passport control before) then go downstairs to the European level. There you board the gates for European flights. So the walk is short now. From upstairs to downstairs.

Roberto, thanks for the details on flights and connection's in FRA and MUC. It's very helpful to have experience from real travelers. I'll look into details on arrival times in Basel.
And it seems like the connections in FRA which are under 2 hrs. should be OK.
Thanks again.
Jerry