If this individual has this much criminal history and would be a danger to the child, they would be luck if a judge would allow supervised visitation.
This is more of a family law question than DWI. I represented a client with a similar situation in Paterson recently. The NJ case law will award visitation to a natural father in almost every instance. The examples of the conduct that do not preclude visitation are very similar to yours. I would suggest retaining a very good family law attorney. The father will have to pay child support and you should get this Order as soon as possible. Visitation can be supervised and he should be evaluated prior to any visitation Order. If a Court does find that he is a danger to the child visitation can be terminated or conditions established to provide for the safety of the child. In my recent case the father failed to comply with Court ordered evaluations and failed to appear in Court. Visitation was terminated until he complied. This procedure may work in your case as well. Call for a free consultation if you like.
Attorney Cheser is correct. You should retain a family lay attorney at this time petition the court for an order setting forth child support, custody and parenting time issues. The NJ courts rarely terminate one's parenting time with a child, however given the facts you set forth above, the court may order supervised visits only with certain conditions attached as well.
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You need to protect your child and get into court right away for a custody and child support order.
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Although I don't like punting questions here with "get a lawyer", the stakes sound too high not to at least get a consult. Whether you're on the Hunterdon side of Lambertville (probably Judge Mawla, possibly Judge Reed) or Mercer (Judge Massi definitely if you two weren't married), you have a decent, smart judge who is generally pretty good and careful.
The issue will be one of proofs. Assuming he'll deny what you're saying, you'll have to make a showing to a judge that your concerns are valid and then request an evaluation and probably supervised parenting time until he shows he's addressed the issues. Your certification explaining it could be enough, but if you have documents (DUI reports, jail report, etc) it'd be a stronger case.
The above is said without seeing your case file and without my understanding the entirety of the facts of your case. Depending on those facts, the above information be may incomplete or may be completely inaccurate. The above is intended as general information only based on what you described and not as legal advice. I advise you to consult with counsel who may be able to provide better information commensurate with a better understanding of your situation.