You can't do better than Bruderholz (Stucki -now deceased, new chef, new decor) in the close in Basel suburbs. Legendary. Recently there for a $400 set menu lunch for two. So you get it all: pricey, the best, a legendary institution and a meal to talk about for years to come.
Coming down a bit in price but not in authentic cuisine or Swiss experience is Braunenmutz on Barfusser Platz for the classic wurst with roesti and beer in again an old style restaurant in the heart of the city. This is a local place, not international in stature but you will see the best of Basel there in an old style beer hall sort of way.
Stay away from the restaurant in Au Violon Hotel which may get a recommendation for its location and uniqueness, but food was totally uninspired in taste though excelled in description. Basel is a very wealthy area, one of the highest per capita income cities in Europe. Many other exceptional restaurants are hidden in the immediate surrounding exurbs in Germany and France.
There is Michelin one star Teufelhof hotel/restaurant in an historic part of Basel which is well worth trying, though it is a bit more quirky and youthful than staid and elegant, and there are very charming local spots down by the river in the old parts of town - St Alben-eck comes to mind as one of the best restaurant choices and there are others in this very lovely ancient St Alban's area.
But number one, you simply cannot go wrong with the legendary Bruderholz. We dropped in for lunch with no reservations but since there is a major convention going on you might want to book now - they have a website. http://www.stuckibasel.ch/index.php?e...
Rehein and Allschwill are two of the nicest suburbs so restaurants, particularly in Rehein might make the grade because that is where much of the money in Basel resides.
The Three Kings as suggested (Drei Konigen/Trois Rois) is a wonderful intown setting right on the river. Close by is Chez Donati I believe is still around - a long time Italian favorite. Right on the river too. Yup, just checked and retains its reputation for elegance and dining in the old traditional style. It gets rave reviews so I would also include this in the top of your list but also one to get reservations for this time ASAP.
Les Trois Rois is renovated and spectacular. Less pricey than the main restaurant is the
Brasserie, overlooking the Rhine. Figure $75/person and well worth it.
Also try the "Bad" (bath) in Basel-Land in Schonenbuch. Great country restaurant
in rural location bordering the Alsace.
The Three Kings is one of the best restaurants in Basel. It was under renovation when I was there in January, but it may be over now.
I had a very good meal at Gasthof Zum Goldenen Sternen (St. Alban-Rheinweg 70).
At the main train station there is a Sprungli shop where you can find great chocolate.
Basel is a lovely city, enjoy it.