Human Health and Disease

Antibodies are produced by B-lymphocytes (which form plasma cells), and T-lymphocytes (helper T-cells) help the B-cells to produce them. If both B- and T-lymphocytes are damaged, no antibodies will be produced against a pathogen. This statement is correct.
Hepatitis B vaccine is produced by recombinant DNA technology in yeast, which is a fungus, not a protozoan.
Snake antivenom contains preformed antibodies, so its injection is passive immunisation, not active.
Injection of dead or inactivated pathogens (vaccination) makes the body produce its own antibodies and memory cells, which is active immunity, not passive.
Hence the correct statement is the one about damage to B- and T-lymphocytes.