Qualifications
The most important qualification for this job is a whole lot of gumption, but research, business, and people skills are also a must. There are art dealing associations you can become a member of, but you don't necessarily have to do that. A go-getter personality and a passion for the job will take you far. That and a private stockpile of priceless ancient knick-knacks.
As with just about any line of employment, a higher degree will assure a potential employer of your experience and know-how. That will increase your value to them, but you would do just as well to study independently and earn some real experience and contacts as early on as possible. Nothing says "hire me" like real-world examples of you bringing in cash.
When you're applying for your first job in the industry, your employer is going to be looking for at least a hint of specialization in something. You're going to eventually need to cultivate a reputation as an expert in at least one area, so be sure you're able to talk about antiques like you know your stuff.