If I hire someone to put up some cabinets for me but they don't have workman's compensation and they are injured on the job, am I responsible?

It really depends upon the reason the accident happened.  Generally, you would not be responsible unless the injury occurred due to some negligence on your part or hazardous condition within your home.  If this is the case, your homeowners insurance could possibly help you with an injury claim.  If you are unsure, I would suggest contacting an attorney and seeking a consultation.