A rule of thumb for the thread depth is use the diameter of the screw. In your case that would be 1/4" for the most strength. Any more would not necessarily make the coupling stronger, but might help get it started during assembly. This would give you 5 threads engaged if you use a 1/4-20. However, I don't believe that this is always necessary. John's suggestion of 3-4 threads would probably hold just fine, especially if there was a good adhesive inside the handle holding it together. If you want more threads engaged, you might go with fine threads instead of coarse, say 1/4-28.
David