As I recall it, sentiment in Kentucky was pretty evenly divided initially, and the state actually tried to remain neutral in the conflict. There was a well-equipped and pro-Southern state militia initially, but as sentiment in the legislature swung against succession that militia was disbanded and replaced by a pro-Union one.
On public sentiment: In the border states, and indeed in some of the non-border states, some individuals and organizations were strong advocates of one side or the other, but there was also a rather large part of the population with no strong commitment to one side or the other, and who would fight with one side or the other or try to avoid the fight based on their perception of their personal interests and their perception of the power/legitimacy of the two sides.
The Lincoln administration had a tightrope to walk in the border states. These were states that had long considered themselves culturally and politically part of the south. Leaders in those states had to fear being politically isolated and powerless in a truncated Union if the southern states successfully left the Union. Most importantly, leaders in the border south had a number of dilemmas: (1) Any war was going to be fought primarily on their territory, and (2) If they stayed in the Union, they would be required to send troops to invade the territories of not just fellow Americans but fellow southerners, and finally (3) They would be doing all of this for an administration most of them didn't vote for and actively despised.
Whatever your other views on the Civil War, it is difficult not to see the degree of Union support in the border south as surprising and politically difficult to pull off. Any major political misstep on the part of the Lincoln administration either in the direction of looking too weak or of looking to eager to war on fellow Americans could quite possibly have swung more border state sentiment toward the confederacy. Lincoln's success on that tightrope was probably the most crucial aspect of Union victory in the war, but it was by no means a sure thing. It took a lot of political skill.