Think this one has done the rounds before (may have even started a thread on it myself).
I do remember reading a Guinness book of Records about the closest and furthest airports, and GVA was mentioned as the closest. I have walked there, but it still took me abut 50 minutes, and I got a lecture at the check in desk for arriving 40 seconds before it closed. "Mr J, vous devez completer le check in avant de fermeture"!
However, there are plenty of smaller airports, especially on islands - NEV is an easy walk to nearby beaches, first habitation up the hill from SAB overlooks the aircraft carrier strip, and of course given the reasons for most people here for visiting SXM, Maho is beach is naturally not far from the terminal!
If we're walking to a concert or exhibition, let's not forget BHX too.
BLL's Lego airport is very close, but I once went out the terminal and onto the awaiting bus, forgetting that my hotel was actually the one inside the terminal!
A taxi driver tried to charge me €50 to go from said leaning tower to PSA control tower, but I got the bus in the end, took about 5 minutes.
You could also ask which Ryanairports are closer to the city than the "proper" airport - that would include CIA, GSE and also IEV, although Ryanair don't go there yet.
The closest of the intercontinental major hub airports must surely be FRA?