The union drafted soldiers in Kentucky. But they rarely if ever got their quota. They asked for 16,000 and got 2000. They asked for 12,000 and got 4,000. Kentucky stayed in the Union (though the Confedetate invasion didn't help) due to slavery. The thought among unionists was slavery is best protected by maintaining the status quo (staying in the union). Joining a risky confederacy meant certain death to slavery if it lost the war. This is why Kentucky had slavery until almost 1866. They tried to prolong slavery for as long as possible. In 1864, union support in the state rapildy diminished. The union treated the state as conquered land. Due to the state being southern and a large number of the young men of the state enlisting in the confederate army. Martial law was established in 1864, confederate supporters were shot in the streets if a Union soldier was killed in the state. This angered the common man. Many Kentuckian's slaves simply ran away and joined the union army. This angered the slave holding class.
A lot of Kentuckians, something like 70 percent of the fighting men, simply sat the war out. Deciding to fight against their fellow southerners or their country was too much for many. And really, the state never made an official choice. Both CSA and USA governments existed with citizens loyal to one or the other. One could make arguments that neither the Union or Confederate governments were represenative.