The easiest way to do this is to have Script B leave some kind of marker when it's running, like a locked file in a pre-arranged location. When it starts up, it write-locks the file, and when it shuts down it unlocks it. On UNIX, this works well, because if Script B crashes the OS will automatically revoke its locks; not sure about Windows.

When Script A wants to check if B is running, it attempts nonblockingly to get a read-lock on this file. If that succeeds, B isn't running, and A should give up the lock. If it doesn't succeed, B is running.

On UNIX, you get locks with fcntl; not sure about Windows.