It doesn’t matter the input size, it hashes down to the same length. It does increase the CPU time, but not the storage space. If the hashing is done on the client side (pre-transmission), then the server has no extra cost.
For example, the hash of a Linux ISO isn’t 10 pages long. If you SHA-256 something, it always results in 256 bits of output.
On the other hand, base 64-ing something does get longer as the input grows.
Hashing is more about obscuring the password if the database gets compromised. I guess they could send 2^256 or 2^512 passwords guesses, but at that point you probably have bigger issues.