How can I see the history of a customer's wallet balance?

Open Plugins → Wallet and go to the Customer Wallets tab.

1. Search by name or email.
2. Click Manage on the customer.
3. Under Recent transactions you see each entry's date and time, type, amount (credits in green, debits in red) and the balance after it.

The customer sees the same list under My Wallet in their account, so you are both looking at the same history.