How can I cut popsicle craft sticks without splitting them?
Craft sticks and Popsicle sticks are not known for their quality when it comes to grain and warping. So when you are cutting them, the stress is released in the easiest way possible which is along the grain. This is especially true because of how thin they are.
One of the following methods should work. They are both related in that I am suggesting multiple cuts.
Scoring
Instead of trying to do one cut you should try and score your sticks first. Scoring will help guide the break and prevent it from going through the rest of the stick. So, score it once then do the rest of the cut.
Multiple shallow cuts
Essentially scoring all the way though. The slower safer method is cut though the sticks in several successive cuts. Use a ruler to help cut straight so you can focus more on your depth of cut. Put more sticks under the ruler to help keep it flat.

Either way you need a thin and sharp blade. You want to cut the fibres of the sticks and not push them apart. So something like standard scissors would be a bad choice.