I recommend Antoines. The menu is in French, and it has been around since 1840. It is not terribly expensive, but you must wear a suit and tie. It is fun watching the tourists show up in a polo shirt and get turned away. Also, it is not pretensious like Europe. They will give you a sport coat and tie but only if you have a dress shirt. But I like having to dress up for once. Oh, and do make a reservation first.
You do not need to dress up to get in to Hermes Bar at Antoines. My wife and I lived in the Quarter for some 20 years and routinely bopped in there for drinks and dinner. Often we were in jeans and t-short (at east I was).