I have a web application that comprises the following:
Here's one way to look at it. Which component should make the decision about which database to use? It's possible that the database (or at least the connection string) could change in the future. Does the website decide which database to use? Or, does the DAL decide?
If you have dev, QA, UAT and prod databases, managing these connection strings is crucial.
If the website decides, it should pass the connection string from its web.config to the DAL. If the website isn't supposed to know or care where the data comes from, then the connection string belongs in the DAL.