This is phrased as a request rather than a policy for a simple reason - We can neither demand nor enforce that anyone do anything in regards to votes. As such, the only solution is to plead. And this, I will do - I would appeal to the better angels of your nature, if you choose to anonymously downvote.
It has been said that any fool can knock over a building. It takes a craftsman to build one up. The one who has labored to create quality content has built something of value (well, most of the time anyway!) and to simply come by and say "That's dumb" has a psychological hurt value.
Downvotes de-incentivize productive participation. Period.
A downvote in any event teaches the one who is engaged not to do what he or she is doing. Sometimes, that may be warranted - if someone is contributing things that are wrong, unclear, off-topic, or just plain not what the site is about (and you'd be surprised how little that actually is!), then correction is warranted.
But - and this is the big but - if you don't leave an explanation, you are unlikely to teach the right lesson.
Lets face it - an anonymous downvote says "You - Shut up!"
Or, the poster might feel like a martyr. "You just disgaree with me. Well, you're just stupid!" Of such things nothing good will come.
A downvote with explanation, however, teaches people with zeal to be better.
So, it's really simple. Love this site? Downvote Publically. Be willing to stick your neck out and actually help. Cowards knock down without building up.
Can I force you to do this? No. But if you love this site, you will follow this commandment.