There are too many issues involved in your case to give you a complete answer here. If your boyfriend is currently in Mexico, there are two ways to bring him in legally: fiancé visa, or as a spouse (you obviously will need to get married first). However, there might be different grounds of inadmissibility that need to be addressed here, such as criminal record and possibly unlawful presence. Talk to a lawyer, preferably - several lawyers, before you take any action.
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It sounds like there might be a couple things for which you would want to consult an attorney. You mentioned that he is your boyfriend. Do the two of you intend to marry? You have different options depending on whether you are planning to marry your boyfriend.
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I think you and your boyfriend need to discuss this with counsel. The criminal issue is likely benign as it was a juvenile action. Owing money is not generally a bar to getting status. And based on your facts it appears he was admitted to the U.S. meaning he can get status much more easily than most. You'll want to ensure all these factors are accurate and then identify a course of action accordingly.