Base64 encodes each set of three bytes into four bytes. In addition the output is padded to always be a multiple of four.
This means that the size of the base-64 representation of a string of size n is:
ceil(n / 3) * 4
So, for a 16kB array, the base-64 representation will be ceil(16*1024/3)*4 = 21848 bytes long ~= 21.8kB.
A rough approximation would be that the size of the data is increased to 4/3 of the original.