I think we can assume that while it isn't really shown or directly referenced there is a lot of stuff being done automatically by a variety of computer systems (in universe). There would have to be for a starship to be truly functional. Kirk mentions (in "Wolf In The Fold") that the computer runs the ship, but I think there has to be be more than one at work. It's possible the reference could be taken as referring to the main computer which exhibits a limited form of artificial intelligence as opposed to a more advanced form as seen with HAL in 2001.
Additionally, they had no concept of computer networks in the 1960s. The idea of saying "computers" plural was probably not even considered. It'd be like saying "brains" plural run a single human body.