What task can the Accounts Receivable (AR) Department engage in to verify that all customers' checks have been appropriately deposited and recorded?

This is a question that has plagued business owners since the beginning of time! Accurate recording of payments is critical to the cash flow of a business and is critical to maintaining a positive relationship with customers as well as vendors.
The volume of transactions paid by checks has decreased over the years. Many organizations and individuals pay by direct transfer to the bank. However, this has not lessened the need for accurate recording of customer checks. Depending on the volume of checks you receive determines the type of system you will want to implement. For example, if you are a small boutique retailer, you probably have more transactions paid by credit or debit card than a check. In this situation, the easiest process is to make copies of checks you receive and reconcile the checks to your deposit slip listing information on the deposit slip specifically to the check. As a further check, daily deposits should be reconciled to the bank statement weekly. It goes without saying every transaction recorded should have a back-up to accurately reflect the transactions of the day and to accurately reconcile to other systems in place.
One way to avoid the hassle of copying checks is if you are a customer of a financial institution that allows you to scan the checks, scan the checks as you receive them. Scanned checks deposit instantly into an account and you eliminate the extra time to copy checks as you retain the original check. Many point of sale cash register systems has the scan feature built into the software eliminating the need for a separate check scan or copy.
If you are large volume check receiver, the problem is exacerbated by the sheer number of checks and transactions that have to be manually copied and recorded. If it is a routine customer, someone you do business with daily or weekly, you can suggest to the customer to send payment by direct deposit to your account. Some businesses use intermediaries like Pay Pal or their financial institution to avoid paper checks altogether. Some businesses offer discounts to payers who send the payment electronically.
If direct payment is not feasible, then, unfortunately, you are back to recording each check in some type of manual or electronic scan ledger. Daily you will reconcile the check to the deposit. Weekly you reconcile the deposit to the bank statement.
Finally, think about the customer. Why are they writing checks and using debit or credit cards? Part of the AR process is not only to record the transactions accurately but to gain quick access to the funds to operate your business. A good account manager will continually review transaction history of customers to find out why they pay with a paper check. Once you understand why then it is incumbent on the business owner to work with customers to reduce the number of transactions from checks required to keep track.