The ability to oppose the closure of a question already exists—users with the privilege to close questions may vote to reopen. Even when moderators vote to close, 5 users can vote to reopen, if they decide that the question was improperly closed.
I don't see any value in adding a 'downvote' mechanism for users to express disagreement with a moderation decision:
If a question is improperly closed, trusted users on the site can vote to reopen. Once there is one reopen vote, the post will be sent to a review queue for users to decide whether to 'Leave Closed' or 'Reopen', so eventually the post will be handled by the community.
Users who don't yet have the privilege to reopen can create a question on the per-site meta for Philosophy, making a clear case for why the question should be reopened. You can see this in action on The Workplace's meta site.
The vast majority of moderator actions can be reviewed and counteracted by the community—closed posts can be reopened by the community; deleted posts can be seen by high-rep users and flagged for review if necessary. Other moderation decisions, such as suspensions, are overseen by Community Managers (employees at Stack Exchange). You can be sure that almost every decision made by a moderator is open for scrutiny by at least one other party, to ensure that moderation is fair and consistent.
Note that the moderator who closed the question left some comments explaining exactly why the question was closed:
Your title question "Does the internet exist?" is too broad to be reasonably answered here. The question in the body of your post, "Would ... anti-realists say [the internet] exists?" is acceptable in scope, but you don't provide us any context or definitions so it remains unanswerable. 1st, define "the internet". You seem to have defined existence as something you can "smell, touch, taste, or hear" but are you sure that definition is satisfactory? Reality, conceptual existence, these are complicated issues that can easily become more confusing if you don't stop to establish a starting point.
Philosophy is a tricky subject, and we don't want to discourage new users from posting questions they have, but we really do want to encourage people to think about their questions themselves before they ask, and write down what they think and what is going on in their minds, too. This way we can work through the problem rather than just guessing what your question is really about and potentially giving you an answer to the wrong question. :)
The issue here is that although you can tell what the question is trying to ask, there isn't sufficient information to give a proper answer on the Philosophy site.
Keep in mind that unclear what you're asking doesn't necessarily mean unintelligible, just that there isn't enough information to reasonably answer in the question's current state. It's also quite broad at the moment, so it could also be closed as too broad.