Biarritz (BIQ) is at walking distance (35 mns) from the city center and the "Grand Plage". The airport is actually closer to the city center than the railway station !
Now if we were to do airports which are closer to town than the railway station, then there might be a few of those - assuming the town has a railway station.
I'll start with SKB - it might only be a sugar train, but the "terminus" is adjacent to the RBIA terminal, both being about 40 mins from Basseterre.