Yes! Under the View Orders tab in the Business Settings > Packages & Gift Certificates page all orders purchased from your store can be viewed. By clicking on a particular client's order you can see how many appointments were used and how many are remaining on the client's code.
You can also find this within the client's History by going to Appointments > Client List then searching for the client.
Clients can track the balance of their package by viewing the confirmation page of the last appointment they booked with the package (they can access their confirmation page from the "Change/Cancel Appointment" link in a confirmation email). Also whenever they try to use the package they'll see the balance, or if it is fully used. If a client registered for an account then any available packages associated with their account will show up when they click "Redeem Coupon/Package" on the scheduling page.