When I joined the Navy in 1989, almost half of my bootcamp company were either ex-convicts or plea-bargain types. One of my closer friends was talked into lying about the deal by a recruiter, which later got him kicked out for lying. He only lied so he could get into the sonar program which required Secret clearance. Most in my company did not lie about their circumstances and the Navy happliy used them to clean toilets and paint ships (airman/seaman apprentices with no job specialty).
In recent years, recruiters have grown even more desperate. High school diploma equivalency was required when I joined, but is not required any more, at least as of 2000 when my friend was a recruiter in Kansas bringing in anyone that could say their name and sign the forms.
What the Navy officially accepts and what they actually recruit and keep are very often two different standards.
I am glad that the submarine community is a little more picky. The worst submarine crews seem like intelligent, high-class snobs compared to typical surface fleet crews. I have lived in barracks where the majority of people there lived like animals.
I can't speak for the other branches, but I am quite certain that it is possible for a potential convict to make a deal to join the Navy, especially if the recruiter needs to meet his quota for the month and only has a few days left.