Yes, your son's criminal past will impact the application, the authorities may ask you all to visit the island for an interview to discuss it.
How can my son who has a criminal record apply for St. Kitts and Nevis citizenship as my dependent?
My son has a criminal record that involved robbery six years ago. He was in prison for half a year. My family wants to obtain passports from St. Kitts and Nevis through its citizenship-by-investment program. I will be the principal applicant, with my wife and son as dependents. Could my son’s criminal record impact his ability to get a St. Kitts and Nevis passport?