After having watched Todd Haynes' wonderful Carol (2015) - adapted from Patricia Highsmith's groundbreaking book of the time, The Price of Salt (1952) - I think he would be a perfect director for the job. Having an utterly brilliant insight and grasp of period pieces and handling certain tabooed topics with great sensitivity, Haynes would certainly master an adaptation of Lehmann's work in my mind.
But also Joe Wright with his great sense of detail and observant (camera) eye would be able to catch the tone and mood of the novel - as he did so beautifully with the 2007 adaptation of Ian McEwan's Atonement (2001) (which actually references Dusty Answer at some point).
Andrea Arnold is another favorite director of mine and in films such as Fish Tank (2009) and Wuthering Heights (2011) she conveys the troubles and ponderings of the youth in a very tactile, sensual and curious way that would mirror Lehmann's language extremely well.
I would also love to see James Kent give it a try - especially following his success of patiently adapting Vera Brittain's memoir Testament of Youth (1933) onto the screen in 2014. Absolutely breathtaking and I am already a big fan!