They do that in London too, special shuttles that run between Gatwick, Heathrow and the other airport I've never seen. My last visit I took the 6 AM train from the town where I was staying over to Gatwick, then the hour-long shuttle to Heathrow for my flight out. For FAR less than the $100+ a taxi would have cost.
Other options include trains. From several airports in the area I could possibly take a train into Strasbourg. Or renting a car -- expensive but sometimes necessary.
Sometimes all the options make it hard. There are 5 or 6 airports within 70 miles of Strasbourg!
Note to self: get cellphone unlocked so I can leave the German cellphone home.