Yes, Gatwick does. For some hotels it is free. For others you must pay the driver. We arrived fairly late in the evening and were staying at the Travel Inn Gatwick. We had a longish wait for the bus (had to pay) and found that the hotel was just a few minutes walk! We walked back to Gatwick the next day.

Here's the URL for Gatwick: http://www.connexionseurope.co.uk/shuttlebus.htm

Frances Toronto, Canada