"growing apprehensive" is correct and usable in written English.
You can use it to describe a feeling of fear or worry that is increasing. For example, "As the storm approached, the campers grew increasingly apprehensive."
There is evidence that the public is growing apprehensive of the government's motives in the far-flung Bay Area Laboratory Co-operative doping case.
Wall Street is growing apprehensive that this spring's green shoots aren't developing into a sustainable recovery, and fearing a deeper retrenchment for the careworn consumer.
Growing apprehensive of the power of the Mughal emperor Humayun, Sultan Bahadur Shah of the Gujarat Sultanate was obliged to sign the Treaty of Bassein with the Portuguese Empire on 23 December 1534.
"This is not the time to go wobbly," Ryan told ABC News' Jonathan Karl when asked about some Republicans growing apprehensive that their support for his plan could hurt their reelection hopes in 2012.
But Cherne had grown apprehensive after the revolution.
Her smile disappeared, and her face grew apprehensive.
Immigrants have grown apprehensive about calling local law enforcement agencies if they know the Border Patrol will respond, he said.
The arrests brought a sigh of relief to residents of this bustling capital city of 11 million people, who had grown apprehensive in recent weeks as the government appeared helpless to solve the bomb attacks.
Just when you start to grow apprehensive, ol' Bobby comes through with the goods: in this case, a blue-chip recipe for glazed Porkette steaks with watercress and black-pepper vinaigrette.
While others cheer Mubarak's fall, Israel grows apprehensive.
Sultan Bahadur Shah had grown apprehensive of the power of Humayun.
