I never really played with it but some time ago I downloaded BGOnline on my iPhone, the app that ipdentifies itself as PhoneUser.MaybeFlaky.net on FIBS. My information is therefore maybe wrong because I don't know it good enough.
I have actually not seen any information about my own rating in the user interface, leave alone what RepBotNG is thinking about me. The screenshots available online also don't suggest anything contrary. So, isn't it possible that many of these evil "droppers" are actually not even aware of the consequences of their behavior? Imagine somebody who downloaded that app for € 1.79 from AppStore, now plays a couple of matches "online" and then comes into a low-equity position and says "F.* it!", and turns off the phone. I somewhat doubt that these people really deem themselves overly evil. It's just a natural reaction: I play against a computer, I'm losing, I click "Cancel", "X", Alt-F4, and I go on my business. Honestly, start playing any other game than backgammon against a computer, and this is exactly your behavior, when you are not aware that your opponent is a human being that you are socializing with in that very moment.
Actually, in a non-professional environment, it is even quite a natural reaction in live-play. Play any odd game against somebody who is not taking this particular game too serious, and maneuver this person into a losing position. If it takes a minute to lose, many people will agree to play to the bitter end. If it takes longer, then the losing party may find it ridiculous to play that farce to an end, maybe because this party is ignorant of more or less marginal winning chances.
@the authors of the mobile apps: Can't you make it more obvious that your users are playing a human, social being? If the rating of the users of the "classic" clients is such a big thing for them, can't you make this rating more obvious? If you see yourself how much it hurts being dropped you are more reluctant to drop yourself, and more willing to continue dropped games.
NOT @the authors of the mobile apps: I think it is mostly impossible to prevent the drop clicks by software. For example in the iphone this is not possible by design. Everybody should therefore find a way to live with that situation. It is not that difficult; dropper-whining can only be topped by dice-whining. Just don't play droppers (there's many ways to avoid them).
If you are not sure about my point, here it is: FIBS today would probably run dry without mobile users. If these people that contribute a good deal to the mere presence of FIBS today misbehave, don't blame them, they maybe don't even hear you. Rather cooperate with the authors of the mobile software, so that they can try to make the problem more apparent to their users. And to the rest of yous: If you are not in the top 100 on FIBS, it is not because of droppers. It is because you are a lousy backgammon player! Like me! ;-)