My son is unemployed and living at home. He has no insurance. I'm supporting him, but worry about bills he could incur.
No, he is adult over the age of majority. You are not liable for debts he incurs unless of course you cosign or otherwise guarantee the debt.
It appears that you son does not qualify as a dependent child. You may able to claim him as a qualifying relative if his gross income is less than his exemption amount ($3650 in 2009) and you provided more than 1/2 of his support for the entire year. Don't forget that unemployment counts as income (although a new provision in 2009 excludes the first $2400).
Please keep in mind that there are many variables that could be applied to the brief facts you have given. The information offered here is general in nature and is not a legal opinion nor is it specific tax advice.
Claiming a dependent does not cause you to become responsible for an adults debts. You would need to accept this legal obligation by co-signing or guaranteeing the debt.
Any individual seeking legal advice for their own situation should retain their own legal counsel as this response provides information that is general in nature and not specific to any person's unique situation. Circular 230 Disclaimer - Advice given in this response cannot be used to eliminate penalties with the IRS or any other governmental agency.