Air France, Swiss, Lufthansa, Jet Airways, British Airways are the main carriers that will take you to Basel from India (Mumbai, Delhi, etc.) You will have one or two stops in between.
EuroAirport Basel Mulhouse Freiburg is a tri-national airport - the only one in the world. It is just 4 km away from Basel. Swiss, Air France, Lufthansa, British Airways, EasyJet are the reputed carriers that arrive and take off from this airport. From the airport, you can either rent a car or take a bus to get you to Basel.
Driving into Basel is also possible. However, it is important to make sure that you have the vignettes for the same. Vignettes are toll stickers, available at the border towns. It costs CHF 40. Getting a taxi from, for eg. Zurich, for four people with luggage, ¾would cost you almost CHF 200.¾
There are two main railway stations in Basel. One is the Basel SBB Station while the other is the Basel Badischer Bahnhof. At Basel SBB, you can expect to find all the trains. At Badischer Bahnhof however, Deutsche Bahn trains can also be spotted arriving. Trains to SBB leave from Zurich, Berne, Geneva. Deutsche Bahn gives you trains running at a faster speed, which leave from Berlin and Frankfurt. Night trains are also operative here, so you do not have to worry if you are getting here late.¾
Buses ¾leave from Zurich, Berne, Geneva and other nearby stops. Zurich's HB stop is where you can find a bus to Basel. Buses can also get you to Basel from Amsterdam and Paris. As mentioned earlier, Basel is extremely well connected and conveniently placed.
You can alight at Basel on the River Rhine. Summer cruise ships are available from Amsterdam to Basel, with stops in between, provided you are fine with the same.