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).
Robin Hood Airport (DSA) near Doncaster is also within 90 minutes and has a good range of destinations from low-cost airlines. Watch out for outrageous car parking charges on short stays (£16 for 12-24 hours, which only drops to £14 even if you book ahead). Longer stays are more in line with other "secondary" airports. This airport replaces the old Sheffield Airport which was next to the M1.
National Express run a coach service from Chesterfield direct to Luton, Heathrow and Gatwick airports about five times a day