One way to spray doors and not make as much of a mess is to start by running 2 screws into the top of the doors near the outside edges and 2 into the bottom, also near the outer edges. Don't run the screws in far; they are just for handles. Then you set the door up on saw horses, on the screws only, on each end. You can put a cheap blue plastic tarp under the whole area.
Proceed with your method of spraying and when one side is done, you can flip the door by the screws, which act as handles. When you have both sides done, you can set the door up against a wall or other surface with just the screw heads touching that surface. You probably will need two people to do this unless you are on the tall side. That's about the neatest way that I know. When you are done, just remove the screws. Since the screw holes are on the edges, you don't even have to fill them.
Tom is right in suggesting that a spray of some type is much better for paint application. From my experience, you get a much better job, no matter how neat you try to be, because the paint does tend to roll down the spaces between the slats. This can so easily result in drips and drops on the back side or whichever becomes the last side done that it is an excercize in total frustration.