Yes, they definitely will. When my husband and I got married, he tried to add me to his checking account, and since I had an unpaid negative balance with a previous bank, they couldn't add me. They did give me a paper with a phone number on it to call, to get things straighted out. As soon as I paid that debt, I got a letter stating it was paid in full from my old bank, and then my husband was able to add me with no problems at all.
The only thing you can do is take her to your bank and let them run the check on her credit - they might not care about NSF checks, as long as she paid her bank.
And not extending credit to her has nothing to do with a credit check - there are other reasons to do a check. In this case, they are doing a lot of things, such as checking the terrorism database to make sure she doesn't have terrorist background. They check to make sure she isn't into fraud - because, as a person who will be able to write checks to your account, she could feasibly write hundreds of checks out of your account for thousands of dollars of stuff - and then disappear, leaving the bank with an insanely negative balance from all the fees. Of course, this would put YOU in a bind, as well - and although I'm sure you don't think she'd do such a thing, it is possible. Even if you believe in her, the bank will watch out for your - and their - best interests.