The commonly used word is
Which carries a contextual meaning to depreciate the item in question relative to its peers/etc. The frame of your question is about "punishing" a conversational contribution - and such "punishment" requires authoritative privileges. These privileges define the role of the Moderator in a conversation.
However, 'to moderate' is not a universal 'downvote,' as moderating can raise the level of attention. Regardless, the common expression for downvoting (and similar) is to moderate.
A commonly related concept is
as in, conversation muting. "Masking" is also used in a similar way.
Where these are words that, in their native contexts (audio, print/art, painting), have an all-or-nothing meaning (e.g. "mute" means "off") – in the context of web media and conversational streams, the words can be used with many shades of meaning. This position is nouveau, and you probably won't find it in a dictionary of any sort (besides EL&U!)
As most are alluding to, the more common figure of speech is given for the emphasizing / highlighting focus of top-ranked elements.