I have little trouble with English in practice, but can’t explain it in grammatical terms, so I’m sympathetic to those who don’t get it. I’m guessing that they see the word “gerund”, and skip over it as excessively technical. Tell ‘em it’s a noun, and they may think [sub]it don’t look like no friggin’ noun[/sub]
I’m not sure this helps:
WTF? Does that mean you can say “We appreciate you not smoking”?
And native English speakers: be careful about trusting your ears.
Trusting your ears is great advice for daily understanding of grammar. The best definition I ever heard was “a grammatical structure is one which a native speaker of the language would use and would agree sounds correct.”
Here’s where people get in trouble with that: they say things like “I hear people saying ‘ain’t’ all the time and I know what it means, therefore it’s grammatical.” Sure, on a certain level. It’s grammatical for very informal uses. As a rule, when someone asks whether something is “grammatical” or not, they don’t want to know whether it’s grammatical for very informal uses, they want to know whether it’s grammatical for standard (i.e., relatively formal) use.
So, yes, if you are a native speaker, you can trust your ears. Until you get into something that confuses you and makes you worry you’re not saying it correctly. Then you should trust the rules of grammar, such as they are.
Good point, phartizan!
Oh, and no, it does not mean you can say “We appreciate you not smoking,” which is still an ungrammatical structure. I think what bartleby.com is saying is that in certain structures, using the nominative (“you”) instead of the genitive (“your”) does not create an ungrammatical structure and thus is equally acceptable. I say, aren’t there enough exceptions in grammar already? 
According to the first link, most grammarians didn’t accept a sentence where the negative not intervened between the pronoun and the gerund.
I’m ain’t so sure about trustin’ you’re ear’s. 
And if you’re not careful, you may label yourself as uneducated. It depends on the context.
Truer words were never spoke.
As I understood it, the direct object in such sentences can be either (a) a gerund, the -ing form of a verb functioning as a noun, which can be modified by a possessive pronoun (your) functioning as an adjective, or (b) a noun clause (phrase?) in which the gerund functions as verb and can take a subject, i.e., you. The context of the main verb indicates which form would be correct. Therefore, if the action described by the gerund is what is being appreciated, forbidden, etc., then form (a) is correct; if the totality of the noun clause is prohibited (“Children under 18 riding bikes,” where adults riding bikes, children riding horses, or mahouts riding elephants are not specified), then the “you” form is in fact correct.
Almost always, the “your” form with the gerund proper as the direct object best describes the situation, and is the proper form to use.
I agree with KneadToKnow’s argument, but I am certain the word should be ‘objective’, not ‘nominative’. A transitive verb always takes an object. If the sentence in question were ‘They appreciate my not smoking’, the choice would be between ‘me’ and ‘my’, and ‘me’ is indisputably the objective case of ‘I’. The word ‘you’ can be either nominative or objective.
On the effectiveness of the wording I am undecided. If the construction seems strange or fussy to all but a few pedants, then that favors rewording. On the other hand, there is a succinct courtesy in the current wording that would be lost in the other suggestions made so far. For example, ‘Please, no smoking’ does not seem to acknowledge or value – to my mind – that there is a choice, to refrain from smoking. ‘Thank you for not smoking’ is possibly a better wording, especially as it uses simpler words, but it might appear to be presumptuous – it assumes the person is not smoking, whereas the original wording does not, and its directness could work against it: people could say, ‘Well, thank you, buddy! but I’m a-gonna smoke anyhow!’
I trust that the sign designers researched their market. In any case, their wording does help retain a logical form. Surely that’s something!
I meant nominative in the sense of a word performing the function of a noun, even if it is not inherently a noun, not nominative as a case, as opposed to genitive or objective.
In looking this up in a couple of dictionaries, though, I see that I appear to have conflated the two words nominal and substantive.
Good catch, *DopeyDave! Thanks!