Winster is about an hour's drive from Nottingham East Midlands (EMA) airport, and a bit further from Manchester Airport (MAN). Bus 199 runs a decent service from Buxton to Manchester Airport (including some small-hours services).
Birmingham Airport (BHX) is a viable option too - 90 minutes plus by car (depending on time of day), and reasonably convenient by train (around 2 hours from Matlock).
National Express run a coach service from Chesterfield direct to Luton, Heathrow and Gatwick airports about five times a day