The first problem is to determine if your ancestor comes from the city of Basel, or from the Canton called Basel-Stadt, or, conceivably, from the Canton called Basel-Landschaft. The old Canton of Basel was divided in 1833 to form Basel-Stadt and Basel-Landschaft. Basel-Stadt contains the city of Basel plus the municipalities of Bettingen and Riehen.
In the event that your ancestor was in fact born in the Canton of Basel-Stadt, the church records you need should be on the FamilySearch.org web site, confusingly and, in my opinion, incorrectly labeled as "Basel City Church Books". If you browse this collection, the first frame is a menu of the parishes included in the collection. You will find Bettingen and Riehen on the menu, so this is a collection of the church books of the Canton of Basel-Stadt, not only those of the city of Basel. Digging deeper into the background information associated with this collection, I eventually found the German title of the collection, which removes the ambituity: Basel-Stadt Kirchenbücher. I can't imagine any Swiss would confuse "Basel-Stadt", the Canton, with the city of Basel!
The next problem is the name of your ancestor. If you find her listed both as Fischer and as Bohringer (probably this should be Böhringer), it might be that her parents were not married when she was born. Sometimes we find notations in the church or civil registration records that show a child originally under her mother's name, and then a later notation citing a legal decision of paternity, such that the child would henceforth be known under the paternal surname. There are other possible explanations. In addition to the surname, you show the given name as either Elisabeth (this is the German spelling of Elizabeth) or Maria Elizabeth. The longer name is likely to be the "real" one; children were often given several names at baptism, but may have used only one of them during adult life.
Next, if you fail to locate the birth in the images on the FamilySearch.org web site, and you decide to try the civil registration offices for either Basel-Stadt or Basel-Landschaft, you will have to decide which date of birth is the real one! You probably won't get much help from the civil registration offices in trying to guess which date to look for, though with the names of the parents, they might be more lenient.
It is always a good idea to do a very thorough search in the places where your Swiss ancestor lived as an adult, before attempting to make the jump into the Swiss records. Sources such as immigration and naturalization records, death certificates, obituaries, and family correspondence should all be checked. Check any sources you may have overlooked, in hopes of removing some of the ambiguity.
The civil registration office (Zivilstandsamt) for the Canton of Basel-Stadt is at www.zivilstandsamt.bs.ch
. In the Canton of Basel-Landschaft, there are 6 regional offices at Arlesheim, Binningen, Laufen, Liestal, Sissach and Waldenburg. When looking for the civil registration offices in the German-speaking cantons, the most reliable method is to find the official web site of the canton or city, then look for an "A-Z" tab or a site map. Then you look for Zivilstandsamt (singular) or Zivilstandsämter (plural) under the letter Z. Each canton manages its own civil registration, so the process used by any particular office may vary.
Good luck with your search!