The best short answer is: by being adults who take responsibility for how we act and how we treat these kids. It's not about coddling or teaching them, it's about us holding up our end of the bargain, even if they aren't. Remember, this topic is about fixing the problem, and if you don't want it to improve, you're in the wrong place. There's a TL;DR for the method I suggest at the bottom, but this method will lead to not only an improved individual raid, but a better kid and community:
Talk to them.
Tell them what the problem is, and ask them to focus and stop talking about the needless stuff.
If they argue and say that other people are chatting about complete BS too, you can observe that the other people are not messing up in the game. But do so kindly, these are KIDS. Make SURE that you give them a chance to adjust their playing by talking to them.
If they ultimately can't do it (adjust their talking or complete the raid requirements), ASK them to leave the fireteam. DO NOT kick them without talking to them. If they refuse or start to cry, explain the reason why you're asking them to leave, KINDLY again, and then ask them again to leave. If they refuse again, then TELL them that you are going to remove them so that people can continue, and only then should you kick them.
You may wish to give them advice on where to find a good guide for that section of the raid so that next time they try they can do better, or to look for someone who knows how to carry instead of a regular raid group. Other forms of encouragement are also a good idea, and not only will they help the child, but they'll leave YOUR fireteam with a much better sense of morale for completing the raid than having just messed up a child.
If you're not a kid, you're an adult (even if you're a YOUNG adult), and you should behave like one where a kid is concerned. IF we keep treating them like crap, then that's the future of our community. If we treat them with respect, they will LEARN respect, both for the game and for their fellow players. And they will LEARN to focus and communicate better. Especially the ones who are taught kindly, because there are no excuses that they can make for themselves when you are honest with them.
If you kick a child without talking to them, they will blame you for their failures, and they wont improve. And neither will our community. So if you complain about our LFG system, THIS is where we start to make a change, by having respect for CHILDREN. Remember, this is a video game, and it's NOT worth the self esteem and emotional wellbeing of a child.
There are also good kids that play this game and it's not fair to the ones who can make it through WotM 390 to put them in the same camp as the ones who don't stop talking.
BTW, I'm 26, so this isn't a kid posting because he's upset, this is a younger brother with three AWESOME older brothers that have taught me what it means to play inclusively and kindly.
TL;DR Treat children with respect and talk to them kindly about what is going on and the situation will improve. Don't do anything without talking to them about it first, including kicking them. It's our responsibility as adults who have made the choice to accept them into our fireteams, and the best way for them to learn. And if you don't want to fix the problem... you're in the wrong topic.
Edit: It's not our responsibility to teach kids manners, but that's not what this is doing, it's teaching them the consequences of misbehaving. But It IS our responsibility, having accepted children into our fireteams, to behave like mature adults. OUR MATURITY, OUR RESPONSIBILITY is what this topic is about.
We, as adults, should be behaving with more maturity than that. Because we're NOT children. And THAT'S the point here. If the problem is going to be fixed (which is why the topic is even here, so what did people complaining about parenting children in the comments expect, someone to suggest crucible patches) then it's the ADULTS who have to start the change.
What, do people think Bungie is going to regulate something? They've done all they can with the rating system. Do people think KIDS are going to change from a forum post? If they're the kind that's acting out so badly, they're not going to change because they read something online. So OF COURSE the change has to start with the way adults behave.
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I don't care if someone is a kid or not. If they are not listening to the squad, causing havoc or just screwing around?, kick them and get someone who wants to play.