I think we need to have downvote buttons wherever we have user generated content on the internet.
Showing their disagreement is the leading reason behind downvotes, irrespective of how downvotes are advertised on most websites. Hence, downvotes allow users to show their disagreement with others while expending very low amount of effort, unlike writing a comment/reply.
It is commonly understood that excessive downvoting of answers/questions by new-comers makes websites less welcoming for them and it is true to some extent on sites like StackOverflow.
However, having downvotes can give signals to the same new-comers that their answers are not quite correct. This negative reinforcement can play a crucial role in the initial stages of expertise; its absence can lead to the learning curve flattening out for the non-experts. We show such an effect on StackOverflow, i.e., users with low level of expertise do not gain much from the website. One of the reasons we think it happens is because even though downvotes are allowed on StackOverflow, they are heavily discouraged (they carry a -2 reputation cost) precisely because of this reason.
Diversity in opinions
Only Likes or Thumbs Up will lead people to discover their niche crowd, but it will not be possible to differentiate disagreement with indifference. For example, an opinion like “Earth is Flat” posted on Facebook (which lacks a downvote button) will likely result in only an eye-roll by people who would consider debating the issue futile, but still may get several hundred likes from like-minded people. This would lead the poster to (incorrectly) believe that hundreds of people agreed with him while nobody disagrees.
From a more computational point of view, by looking at up and down votes, we can actually uncover the complexity of opinions and diversity in conversations.
One of the reasons against having downvotes seems to be that it promotes toxicity on platforms by allowing vote brigading. This can also be used as a means of censorship.
While there are isolated incidents of such brigading happening, it is unclear whether it effects enough user content to be an issue. If it does become a significant problem, there are methods which can be used to identify and weed out the effects.