The taking of Kentucky is a must for the Union since it becomes the base for the 1862 campaign to take Tennessee. The thing that makes it so iffy is the amount of luck needed to make it go smoothly. The Union would prefer to wait until October to make the invasion that will violate Kentucky neutrally and possibly force them to become a Confederate aligned state. This isn’t that serious but it makes taking it easier since neutral states don’t generate militia to oppose you. The main reason for waiting for October is the political cost of invading. You lose 150, 100 or 60 Political Points if you strike in July, August or September. Most of the points are made up by the regions you grab but it is still a significant cost. On the other hand moving early prevents the South from getting the jump on you and solves the initiative problem since attacking doesn’t require all that many units to have initiative. If you wait to long and the South decides to beat you to it they take the political hit for it but if the Union fails to get Initiative in their Reaction phase they may not be able to retake the key areas of Paducah and Louisville. The most dangerous time is the September turn when the cost to the South is low and your reaction turn will be in winter. For the Union player waiting for October for invading besides giving the Rebs a chance at a September invasion carries the risk that they will not get Initiative at all. For sure the South will strike in October if the Union has failed to make their move.
The taking of Kentucky is quite simple when unopposed. The Union positions transports and gunboats to support the move the turn before, then moves forces hopefully large enough to overrun the areas wanted. Paducah is easy since the main Union army is in Cairo and usually moves across to seize it. The offensive from Cincinnati is harder since this group is less likely to get initiative. Station cavalry units, as many as you can, here so they can use their extended movement to capture regions by overrun deep into Kentucky then use normal movement to bring up the infantry to occupy them. Amphibious assaults can also be used to get troops into these areas.
In my case I had little initiative so had to rely on cavalry to open the way. The red arrow from Cincinnati shows my initial overrun by cavalry to open the way, then the main infantry movements in dark blue to overrun Louisville, Paducah and Henderson. Due to lack of cavalry I used one amphibious move to seize Bowling Green shown in purple. These attacks were then reinforced by normal and strategic movement of supporting troops. One the following turn the control of Kentucky was expanded by occupation of Ashland and Glascow by cavalry overruns. The final two population centers of Lebanon and London weren’t taken until later due to winter movement problems (green arrows).
Follow up attacks should be able to extend your control across most of Kentucky right up to the Tennessee border. The most likely thing to stop you is winter movement restrictions and lack of Initiative. Most of the rest of winter is spent extending your grip on Kentucky and preparing you army for the spring offensives.
If the South jumps the gun and occupies Kentucky first using they same tactics, deep cavalry intrusions followed by infantry garrisons, it can be a tough nut to crack. How difficult depends on what kind of Initiative you get, whether Kentucky goes Union putting the Rebels at a disadvantage, and the winter penalties. Getting a foothold on the other side of the river is your main objective; that is, taking Paducah and Lexington. You will probably spend the rest of the winter of 1861/1862 driving the Rebels out of Kentucky.
Refer to AAR for more info: 2nd ARR WbtS Whitehead(USA)