Can I marry him and be his sponsor to apply for residence even if he has a criminal history?

Although not completely clear from your question, I presume that you are a U.S. citizen and that your fiance entered the U.S. lawfully and with inspection (showed a valid visa when he entered. In that event, upon becoming married you may sponsor him in an adjustment of status application process by which he will seek to become a Lawful Permanent Resident (get a "Green Card"). As part of that application process, he will need to address his three DUI cases, and depending upon the details he may be required to provide court-certified copies of arrest disposition documents and even may be required to provide evidence that he is not a "habitual drunkard" - an awkward term used by the USCIS in assessment of adjustment of status eligibility. It would be wise for you and your fiance to consult with an immigration attorney who, after learning all the relevant details, could advise about immigration eligibilities, options and strategies, and could offer legal representation in the often complex application process.