Make sure you consult the 12's schedule and have the possible departure times mapped, the buses connect at the White Plains Trans Center, at the Metro-North Station, a large bus hub like those found in the downtown's of most cities (the last stop on the 12), they also take similar routes through in/out of White Plains making some of the same stops going to and from Downtown White Plains so you will be backtracking a bit. I'd tell the bus driver when you board that your transferring to the I-Bus and see if they tell you about a closer stop. (that would require crossing the street).
Bee Line is $2.50 payable with COINS ONLY (or a swipe on a NYC Subway/Bus MetroCard, that are also accepted on Westchester Bee-Line) ask for a paper transfer.
I-Bus (I've linked to its website, its a route that has its completely own) is $2.60 (until 1/18/14) when it will go up to $3, CT Transit that runs the route accepts dollar bills but exact change is required. Show the driver your paper transfer from Bee-Line, it might give you a discount/free. I know Bee-Line and CT Transit have some reciprocity (the I-Bus website says its transfers are acceptable on Bee-Line) but I don't know how I-Bus accepts Bee-Line Transfers.
My advice is to make sure you have 3 dollars in coins on you (golden dollar coins are accepted) and $3 in bills to be completely covered. (more coins if your traveling before 1/18 and I-Bus fare is slightly awkward, dime required $2.60 fare).
(For the historical record: There used to be a direct Bee-Line Bus Shuttle called Airlink from the White Plains TransCenter to the Airport but it was discontinued in 2010, in my mind I thought it still existed).
