Specifically methylates cytosine 38 in the anticodon loop of tRNA(Asp). cytidine(38) in tRNA + S-adenosyl-L-methionine = 5-methylcytidine(38) in tRNA + H(+) + S-adenosyl-L-homocysteine Belongs to the class I-like SAM-binding methyltransferase superfamily. C5-methyltransferase family.