It depends
Whether you realize it or not, your account is considered a bad debt. Although banks do not usually report bad debts to the credit bureaus, they can certainly do so.
From what I know, it depends on the balance. If it’s lower than $300 than most banks typically do not report it to the bureaus. However, if they sell your debt to collection than the collection agency will report it.
One thing for sure – they will report it on the Chexsystem that links to all banks, so when you try to open a new checking account elsewhere you won’t be able to open a new account.
Having a charge-off or a collection account may cost you up to 120 points off your credit score. Not to mention that the bank or the collection agency they sell the debt to can file a law suite, and most likely will get a judgment.
My advice – Call the bank tomorrow and make a commitment to pay those NSF fees off. After that, if you prefer not to have a checking account and use cash instead, then fine, close the account. But get the account closed with zero balance.
BTW, if you don’t want a checking account you can always use money orders to pay for stuff you need checks for. Those cost a little more than regular checks, but far cheaper than paying the bank huge NSF fees each month.