The answer to your question is dependent on a number of things.
First of all, what are the terms of the lease agreement between you and your tenant? Is there a written lease? If so, did it state that the tenant is responsible for utilities? that the landlord is? The responsibility will lie with whomever it was contractually given to.
Secondly, how were the utility accounts set up? If the bills were put in your tenant's name with you as co-signer (or guarantor or surer), then you are considered to share the responsibility for them and the failure to pay them can damage your credit. If they are solely in your tenants name, then you are not. (Your credit can only be affected by the payment or non-payment of bills that are tied to your credit.)