Falsity implies anything . B is only false when A is true but B is false. B is automatically true. This is a definition and the only criteria to establishment of the falsity or veracity of a particular implication however paradoxical it may sound. For example, If you are not reading this sentence then I have not written it.
B appear as a major premise of the modus ponens .
Modus Ponens is one of the syllogisms which are a form of a deductive reasoning. Modus tollens is another. A is the minor premise of the modus ponens. So falsity implies anything. There are some trivial examples.