There is no quote directly stating it one way or another but there is some evidence both for and against.
For
* Dementors "live" on the island of Azkaban. Although the island is nearish to the coast, <PERSON> manages to swim the distance, I doubt the Dementors swim there. Maybe they get taken their by ministry boat or side-along apparition (I wouldn't want to be a Dementor transporter) but they still can clear the distance themselves when they escape Azkaban to attack <PERSON> in OotP. This suggests they can fly but they might only be able to move at low level, just above the water so it isn't conclusive proof of full flight.
* The Dementors manage to go long distances unseen. They do this to reach <PERSON> in OotP and they carry this on when they travel with <PERSON> in later books. Travelling at ground level they would almost certainly be spotted by wizards and/or muggles (muggles may only see Dementors as drifting fog as in the Half Blood Prince but purposefully drifting fog crossing Britain would still generate some news that the Order would pick up on). 821
The Dementors apparently are not seen as if they were <PERSON> would mention it in the trial and the Order would follow the Dementors and track <PERSON> or at least mention the possibility. As no one does mention this it suggests Dementors can travel above ground in the cloud cover where they would be hard to spot.
Against
* When the Demetors come to the Quidditch pitch in the Prisoner of Azkaban it is suggested they were hungry and couldn't resist so much happiness all in one place. If this were the case why would they stand on the ground so far from the happiness rather than fly up to the players or the spectators in the stands. Since the Dementors didn't go closer to the spectators and just stood their watching it suggests they cannot fly.
* No Dementors are around when the Order escort <PERSON> from the <PERSON>'s house in the Deathly Hallows. Dementors would be useful here as they would cause panic among the Order's ranks and allow the Death Eaters to work out who was the real <PERSON> by watching who has a stag patronus. The only reason not to bring Dementors would be if they cannot fly.
* Dementors are often described as fleeing before patronuses (patroni?). If they can fly why do they never attempt to flee from the patroneses via flight? 821