The loss of Fort Douaumont was a humiliation for the French and a public relations disaster. It was partly due to an intense desire to redeem themselves that the French launched such costly and furious counter attacks to get it back. Had it not fallen (and it could have held but for a series of ridiculous mistakes), the Verdun sector would still have seen bitter fighting, but the French might have not gone to the extreme lengths that they did IOTL.
On the other hand, the fall of Douaumont gave the Germans hope that they might actually be able to capture Verdun, which had not been their original plan. Their intent at the outset was merely to create a perceived threat to the city in the minds of the French that they would be able to lure an enormous French force into the sector and destroy it through massed firepower. So perhaps if they had not taken Douaumont, the Germans might have struck to their original plan and been the better for it strategically.