Padding always the same, problem or not?
No, it is not a problem to have the same padding at the end of a message. Some known plaintext is often available in practice (protocol fields, etc). The security of the system is (or at least should be) based entirely on the key.

If you are really worried about it for some reason just change the IV for each 64-byte block. Or as CodesInChaos said in a comment, if messages are always a multiple of 16 bytes, don't even use padding.