Downvotes are not only anonymous, they are actually  invisible, i.e. you don't even know the number of downvotes an answer has got. I hope you are not talkling about that, as that will totally kill the transparency of ordering of answers!

Assuming that you are talking about making them just anonymous, not invisible (i.e. the answer would simply display Upvoted by X Quora members), I will still argue against making them anonymous. I feel that Quora upvotes serve at least the following purposes, and making them anonymous will kill/limit their utility for these purposes. I have also included examples from my own experience with each of the points. 

  • Validation: Most authoritative answers are usually the ones that are either written by subject-matter experts or industry-insiders or upvoted by them. In absence of named upvotes, we will lose the ability to identify the best answers by second of these criteria.
    The Subject matter experts may not have time to answer all the questions from their field of expertise. An upvote is a quick and easy way to establish authenticity of an existing answer without creating a redundant answer. This way named upvotes are helpful to the original questioner, other readers,  the SME and the answerer (See next point)

    As an example: My answer to Who is the oldest LinkedIn member here in Quora? was upvoted by Lee Hower,  one of the LinkedIn founders. After his upvote, anyone reading the answer will have no confusion on whether my answer is correct. (He did leave a comment too, but only his upvote would have sufficed to establish the authenticity of the answer)

  • Encouragement to answer: You may have seen people requesting celebrities to retweet/reply to their tweets on Twitter. Seeing your answer getting upvoted by a power-user is similar, but perhaps 100 times more encouraging! This not only encourages answerers to write more answers, but also sharing these stories with non-Quorans, enticing them to join Quora and thus, making this an even better community.

    Going back to the same example as previous point: My favorite Quora story is not about the answer that got me 60 upvotes; it's about the oldest LinkedIn member on Quora answer getting upvoted by the LinkedIn founder. I have shared this with many (in person and on FB) and I am sure this has made some of my friends join Quora.

  • Virality: Upvotes also serve the purpose similar to that of share on FB. Every time, I click the upvote button, it gets pushed to my followers making the content viral and improving the social aspect of the community. (I totally agree with the concerns about upvoting some answers raised by the Anon user in a previous answer, but I guess an easier solution to that is moving the "Make Anonymous" link from bottom of the page to the top, allowing people to upvote anonymously if they want to.

    For example: 60 upvotes answer that I mentioned above is about 3 months old, but still gets an occasional upvote, followed by about 3-4 more upvotes from the followers of the new upvoter on the same day! In absence of named and shared upvotes, answers like these would usually get buried among numerous others.

  • Transparency: Finally, named upvotes allow me to understand the order of answers. It's not uncommon to see an answer with x votes placed higher than another answer with x+10 votes, even when none of the users is power-user. It is usually because of the first answer getting upvoted by some power-users. I believe this is a very good feature somewhat analogous to search-engines not just counting the number of pages linking to your page, but also the credibility of those pages.
     Without named upvotes, we will either lose this very good feature, or if the ordering algorithm in the back-end does not change, we will lose the transparency in ordering the answer. It would be hard to explain why an answer with 20 upvotes is placed higher than another with 27 upvotes!

    For example: my answer to Behavioral Economics: Why do people pay $8 for a dessert with no second thought but won't buy a 99-cent iPhone/Android app without thinking hard if its worth it? with 22 upvotes is rated below another answer with 18 upvotes. Seemed a bit weird at first, but then, I realized that not only is Stephen Frank a more active Quoran than I am, but also his answer has been upvoted by some power-users like Gil Yehuda, Chris McCoy and Marc Bodnick.

So in sum, I certainly want Named upvotes to stay! Just make anonymous upvoting easy, and I think we should be all right!

See question on Quora