To be sure, I figure that Vicksburg falling was more symbolic than literal as it wouldn't be impossible to slip through, but I would expect that that symbolic loss, with the Union now able to patrol the entire river unmolested by shore batteries or enemy ships, would be a pretty devastating blow not just to morale generally, but most specifically to those now cut off from 'the heart' of the Confederacy.
So how did the public in places like Arkansas or Texas respond to this, and what kind of impact did it have on support for the war: both within the civilian population as well as the military stationed there?
The Trans-Mississippi Confederacy was essentially an orphaned entity after the U.S. closed the Vicksburg/Port Hudson corridor in July 1863. People were pretty easily resigned to it, to be honest. The primary sources I've read all felt like they knew that it was coming, and they knew that it meant they would get no more support from the Confederate government. By this point, they had grown used to feeling abandoned and neglected. In early 1862, a U.S. Army invaded Arkansas from Missouri and trounced the biggest Confederate Army ever assembled west of the Mississippi River. That Confederate Army then completely bailed on Arkansas and crossed the Mississippi to fight U.S. Grant's army instead. The U.S. Army left alone in Arkansas marched pretty much unopposed through the state - as far as their supply lines would allow. They ended up freeing thousands of slaves in the Arkansas Delta and setting up a Union enclave on the Mississippi River port of Helena.
The Confederacy sent Maj. Gen. Thomas C. Hindman, a former U.S. Congressman from Arkansas, back to his state to get things under control. He promptly pissed off EVERYONE by declaring martial law, enforcing conscription, and encouraging people to form guerrilla bands to attack Union soldiers. These bands, of course, ended up killing and robbing indiscriminately. So people weren't feeling very positively toward the Confederacy by mid 1863 anyway.
Hindman actually managed to build an army, which promptly got thrashed at the Battle of Prairie Grove and pretty much melted away on the retreat. A few weeks later, the U.S. captured the largest body of Confederate troops remaining in the state, 5,000 men, at the Battle of Arkansas Post. This was January 1863 and Confederate Arkansas was already kaput.
Little Rock fell in September 1863. The Confederate state government retreated to Washington, in southwest Arkansas - not far from the Louisiana state government in exile at Shreveport and the Missouri Confederate state government which was at, I think, Marshall, TX or somewhere else in the area. The Confederate Trans-Mississippi Department was also headquartered at Shreveport, commanded by Lt. Gen. Edmund Kirby Smith. By late 1863, organized Confederate authority in the Trans-Mississippi had greatly diminished its geographic scale of control. What was left was so isolated that it was referred to as "Kirby Smith's Confederacy."
When Smith was appointed to command the Trans-Mississippi in February 1863, he was bummed because he knew it was a dead-end job. "You might as well bury me," he wrote to Jefferson Davis.
To sum it all up, the fall of Vicksburg fulfilled the long-running fears of people in the Trans-Mississippi, that they were on their own.
Sources:
Jeffrey Prushankin, A Crisis in Confederate Command
William L. Shea, Fields of Blood: The Prairie Grove Campaign
Those will get you started. To be honest, a lot of this is from memory since I'm sort of writing a PhD dissertation on Civil War Arkansas.