It seems that downvotes would definitely decrease - I would bet that a large portion of them would not have been made in public. If nothing else it might increase the number of replies to a post - It would be slightly embarrassing to see your name as one of the downvoters if you didn't also reply to the post.

Enabling downvotes on submissions could also be interesting - maybe to downvote a submission you have to comment about it or about why you downvoted it. Revealing voting could add an additional feature to the site - if we see that someone who likes the same stuff voted something up, then we'll probably read it.