Congratulations on making the sacrifice to be a stay at home mom. It is unclear in your question as to whether you are looking for a home to rent or buy so I will cover both for you.
When purchasing a home, your boyfriend will want to contact a local home lender to get pre-approved and they will provide him with how much they will loan him for a home and how much money would be required down. What your boyfriend is qualified to buy will be based on income, debt, and credit history. With a small baby I would recommend that he purchase a home on the lower end of the approval range so you have money for emergencies and those costly diapers. If your credit is worthy you can still go on the loan with him. If your credit is not worthy then you can go on title which you would want to do so that you would have rights if something happened to your boyfriend.
When renting, most landlords pull your credit reports because they want to know that you are likely to pay rent each month. There are some landlords that go by references and do not check credit but they typically require larger deposits.
My question to you would be where are you living now? If you are paying rent now and barely making it then you would want to make sure that you buy a home with the same payment as what you are currently paying. Live within your means! You will actually be ahead at the end of the year as you will get some tax benefits from owning. Check with your accountant for the benefits.
Congratulations on your new baby!