There is never a reason for a system to have downvotes. The vote of 0 is plenty enough to show that no one cares or thinks little of the question, comment, link, whatever in the system used. Closing, Locking, or Deleting solves what downvotes would otherwise solve as well.
Think about the downvoting of comments on sites like reddit. What other use are they besides to tell others they are [insert favored insult]? They serve no purpose but to insult other users, or in other words: troll them. It is a literal troll feature, developed by a troll designer, and only a troll would implement such a feature to any system. Perhaps if the downvotes flagged something as spam to hide the comments, such as youtube does. However, most downvote systems fail to implement this, and instead have a SEPARATE function for reporting spam or inappropriate comments. Once again, making it useless. Downvoting is also NOT a good implementation of spam/deletion because unpopular opinions that do not violate the rules will become hidden or deleted, which is counter-productive to a free-speech environment. Such a system allows for trolls to abuse others by simply disliking their comments, making them disappear. Imagine if downvotes deleted that which was downvoted. Trolls can already delete any comments they wish on StackExchange, and moderators can do nothing to prevent it or reverse the damage.
Downvotes were made by a troll designer who enjoys spreading negativity on the internet (as if there wasn't enough already...) and every troll after him decided it was a good idea to mimic the design.
Few people actually contemplate whether or not the feature being used is good, bad, useful, effective, etc. They merely copy/paste from design to design and change parts they like or dislike, without any thought to creating their own system.
If we had more originality in web design, and less mindless copy/paste, we would have more people implementing their own systems and not other people's. We would see the elimination of pointless troll features such as downvoting, and an increase in innovative features such as upvoting (which give good statements more visibility). Innovative features that would otherwise not be created or improved upon by original thought in design.
Of course, that is to naively pretend that every web developer is a great web designer. Honestly, it is probably best that people copy what works, due to the high possibility that their lack of skill will produce a worse design (or even a nightmarish one that no good designer wishes to even fathom!) As it stands, all of the good designers ARE original. They are allowed to be original, even encouraged to be so, due to their talent as a good designer. The rest just copy/paste and ruin or improve small bits at a time. It takes an intelligent person to instantly think "Hmm...what is the benefit of a downvoting system? There sure are negatives..."
Downvoting closed answers is simply a redundancy to the stupidity of the downvoting in the first place.