In my A/R aging report it shows several negative customer balances, but when I go to write them off as bad debts, QB tells me there are no open invoices for those customers. How do I resolve this?

Negative balances mean the customer has overpaid compared to what you have billed them.  In in case it would be unusual to have open invoices.  Instead there are likely open/unapplied payments due to the over payment state the customer is in.

If the balance is incorrect, fix it by adding an invoice and then applying the payment to it.

If the balance is correct then there may be nothing to do, or you may want to issue a refund check coded to AR to bring the balance to 0.00.

I am unclear from your post whether you are experiencing negative balances in a/r or overdue invoices.  They are two different things and I will try to explain.

Negative customer balances on the aging report actually represent over payments made by customer or credits which are greater than the outstanding invoices.   If you want to remove the negative balances you must enter an invoice to offset them.  The steps to do that are as follows:

Enter an invoice for the customer and prior to saving & closing the invoice click on the "apply credit" button on the top of the invoice under the "main" heading.  The overpayment / credit should appear and you can place a check next to it to apply it to your new invoice.

Outstanding balances for a customer can be removed by going to the customer center, highlighting the customer in question, clicking on the "transaction" tab, changing the "show" section to invoices, changing the filter to open invoices.  Then click on the invoice you want to charge off the balance.  Under the "main" tab at the top of the invoice choose "refund/credit.  This will allow you to enter a credit directly from the outstanding invoice.  This credit can be changed to be a partial of the original invoice, it does not have to be the entire invoice or even show exactly the same items.  You could for example create an item that is associated with bad debt and use that item on the credit to go against the invoice that is outstanding.  

I hope I have answered your concerns, but if not, please feel free to contact me for further details.

Customers with negative balances are monies you owe to them (refunds).

To close these you are not 'writing them off'- its the opposite, you have to create an invoice to them to absorb the credit.  Yes this will create income in the current year. 

Let's link the deposits to the invoice, Jmp17.

I'll show how to do it:

1. Open the Invoice, then click Receive Payments.
2. In the Customer Payment window, select Apply Credits.
3. Put a check mark in the box beside the Date of the invoice.
4. Then, click Done.

I've added screenshot as reference:

Let me know how these steps works. I'll be here if you need more help.