Well, I have a couple of ideas. But these are just ideas, I don't know how well they will work.
1: Wood dye. Like @Chris H said, but maybe you try it yourself. Get some wood dye, put it in a small bowl, and soak the balls in it, then let them dry.
2: Spray painting. This is a rough idea, but it might work. Take a block of foam, or a sheet, this will be piece A, and lightly press the balls into it so they loosely fit. Spray, then let dry. Then, take another piece (piece b) and press onto the top. Now, together, flip them over, so piece A is on the top, and B is on the bottom. Now pull off A. Now the un-painted side of the balls are up. Spray them, let dry. Clear coat if you want, then flip again and clear coat the other side. Repeat this process. The use of the foam will make it so that you don't have to individually flip each ball one at a time, and it also holds them nicely so that they don't get weird marks from resting on a drying rack.
Those are the two simplest ideas I have that would be the most time efficient. If you go to a certain online ordering site that has the same name as a large rain forest, you can find a pack of 1000 uncolored wooden balls for about $140.00. One thing to keep in mind it that the dye might not be able to soak through a protective coating, and paint might not stick, so try to look for something that's unfinished.