KY joins a side, loyalty will lean toward that side, plus you get more militia and other units, plus IIRC a slightly larger KY unit pool.
I'm not sure if the "too many troops too close to the border" events are still in the game, but I don't thinks so, so other than outright invading KY, you just have to sit back and wait.
IMHO, the longer KY remains neutral, the better... up to a point.
Early in the war, the South is far stronger in the west and far west than the Union, and the Union already has it's hands full with MO.
If KY goes early to the South, and the South is aggressive in TN and KY, she can catch the Union unprepared and do some real damage.
But when the Union has built up forces in the Old North-West, and MO is stable, it will be time to move south, and without KY things are ... interesting. I played one game in which KY remained neutral throughout the war, which meant, attacking up the Tennessee and down the Mississippi to establish bridgeheads. It was an interesting challenge for sure. LOTS of naval action and invasions.
In the end, I don't think it wise for either side to invade KY; the NM cost is simply far too high.
If KY hasn't declared itself by '62, as the Union you better start working on getting around KY, because the clock is ticking. If you are playing the South ... lucky lil' son-of-a-gun aren't you