I inherited a sizeable collection of art and antiques, and a local dealer has offered to purchase them. How do I know if their offer is a fair price?

You need to find an appraiser in the field of antiques and collectibles - not just a "general property" appraiser. There are a number of organizations out there from ISA (International Society of Appraisers), the AIA (Asheford Institute Of Antiques) and ASA (American Society of Appraisers). All of these organizations offer credentialed appraisers, however, I believe the Asheford Institute is the only one that specializes in appraisals of just antiques and collectibles.

If the collection you inherited is sizable, then it's worth paying a professional appraiser to evaluate your items. Your appraisal will include a sliding scale of values - i.e. one for retail (what an antique store might sell it for), one for wholesale (what an antique store might buy it for), an auction value, as well as an insurance value. Once you have this appraisal, you should be able to refer to it, to reasonably ascertain whether or not someone is giving you a legitimate offer for your inherited antiques.