This is the same background check used to screen pilots and airport security personnel.
Having a criminal record is not an immediate cause for denial of a license.
If the conviction happened over one year ago, you shouldn’t have a problem.
We’ve had students with felony convictions, DUIs, etc. get their FAA remote pilot certificate without issue.
At the end of the day, they just want to make sure you’re not on any terrorist lists or already flagged in their system for some kind of serious aviation offense.