I am engaged to get married. I owe back taxes. Will this become my wife's problem too once we are married?

No. It is your problem and she does not inherit it. The only way she could be negatively affected by your past tax problem would be if you file jointly, get a refund, and the IRS recoups that refund to go toward your back taxes, which can happen. Also, make sure that you either pay in taxes if you are self-employed, or are having ample taxes withheld from your pay. Because if you owe and don't pay, and you file jointly after you are married, she will have a tax problem.