I own 50% of a home, which I would like to use to write off the taxes, as I will be paying the bills. The other person invested 50% up front and they would like to rent out a room in the house. If they do the rental contract, do I have any tax responsibilities on that income? Since I would be the primary person paying the loan, can I keep the tax credit without having the responsibility of said income?
Your question doesn't make much sense. You should probably talk to a licensed tax adviser (EA or CPA licensed in your state), with more detailed explanation of the specifics of your situation.
In general, you cannot write off something you own, unless you use it to produce income. So just owning a home doesn't entitle you to any write offs. You mentioned a tax credit, I'm not sure I can think of any relevant tax credit in this context, so I really can't tell what you're talking about.
As to the other person - if he's the one renting, his portion of the house, and he is receiving the income, then it has nothing to do with you.