It depends.... mapquest says it will take a little over an hour, but that's mapquest. It doesn't factor in traffic. You can take I84 to the Taconic, and from the Taconic you can pick up the Sprain Brook to I287 into White Plains.
But be warned: it probably won't take you just one hour if you go that route, because 287 is always a nightmare no matter when you drive it. There is perpetual construction and there is always a ton of traffic. If you take 287 it will tack at least 15 minutes onto your commute, and that's on a good day.
If you'd like to live in Wappingers and commute to White Plains, I'd suggest taking the Taconic down to Route 35 East, which meanders through Yorktown Heights. From there you can pick up I684 to White Plains, which is a better road and will allow you to get to work on time and keep your sanity.
The train situation is a little more complicated. White Plains is on the Harlem line, and the nearest Metro North stations to Wappingers (New Hamburg or Poughkeepsie), are both on the Hudson line. If you wanted to take the train you'd have to drive from Wappingers to the nearest Harlem line station, which I believe would be either Dover Plains or Harlem Valley-Wingdale. But I just garnered that from looking quickly at a map of the stations on the MTA website. (
MTA Home Page)
Either way, if you wanted to take the train to White Plains you'd have to drive from Wappingers to the other side of Dutchess County to get to the nearest station. As an example, the drive from Wappingers to the Dover Plains station is around 45 minutes. Then you'd be on the train for over an hour before you even got to White Plains.
Unless you really prefer the trains, I'd definitely drive from Wappingers to White Plains. Your commute will be over an hour driving, and will be more like 2 hours if you take the trains. Have you considered Putnam County? It's above Westchester but south of Wappingers, and living there might cut down on your commute time - both driving and train-wise.