I suspect that its up to you to offset your emissions which you can do via a number of websites rather than rely on the airline. There is a lot of hype about using biofuels, etc but the amounts are quite low compared with their total total use of fuel. As a form of "pump priming" the airlines are right though.
If you want to minimise your carbon footprint:
Choose a direct routing (although this may not be true if you are occuping seats that would otherwise be empty on connecting flights).
Fly an efficient aircraft type. In order of efficiency (lest efficient first) I would suggest 747, 340, 380, 777/330, 787/350 where slashes indicate I can't tell them apart.
Fly economy and the densest seating available. (I am guilty here, one of the reasons for flying SQ to Oz last year was its 9 abreast seating in their 777s)
Fly at a quiet time when there are seats that would be empty anyway (February for the North Atlantic, off peak times in European business routes, etc). I'm not sure about the Oz route.
Always remember to consider the whole cost of energy. How much energy went into constructing Greta's solar powered boat (which probably had an auxiliary engine anyway). I suspect that the cost of constructing planes is a small proportion of the total energy used. This may not be true for cars.
Don't forget the cost of getting to the airport. You are probably better off giving up driving rather than flying. If you don't drive and find Ebbsfleet Parkway really inconvenient - well fly!
I always say that there are three options with climate change:
Ignore it altogether and see more Australian bush fires. (Think of the great American dust bowl as to what happens when you don't understand the climate
Go down the Greta T route and see mass unemployment, depravation, social disorder, etc.
Try and find a solution may a little bit more for carbon (quite possibly stimulating economic growth so not significantly our wallets. Who knows, we could live happily ever after.
(Sorry I ended up being rather polemical)