Can a botnet amass and coordinate enough computing power to break encryption keys that are typically considered to be secure (through brute force)? If this is the case, do we need to make our encryption keys even stronger (e.g. higher number of bits in the encryption key)?
No, most cryptographic methods that are considered secure are based around assuming every computer in the world is working on the problem non-stop and it still takes to the heat death of the universe.
Sure it could be used to break less secure encryption, and in fact, using a super computer to crack such weak keys is effectively the same thing as using a bot net, but that bit length wouldn't be considered particularly secure if that would make a difference.
It is generally a safe assumption that government entities can put far, far more computing power in to trying to crack something than even the most powerful botnet could handle. And there are far more profitable things for botnets to be working on.