Okay so honestly, the thought of children attending our wedding didn't really occur to me until I read that insanely long (and hilarious) thread on I&P about wording invites. On FI's side, there are about 10 children (all kids of his family members, under drinking age) and on my side, I only have one cousin with a baby and one friend with a baby. FI's brother is also getting married next year and is having no children at their wedding. FI's dad apparently didn't like the idea that they're not having kids at their wedding since there are so many in their family and he brought the idea up to us. Initially, I wasn't really bothered with kids being there, but now I'm not so sure. I asked my mom yesterday and she thinks it's best not to have kids there for a few reasons (which I agree with). I've tried to search the boards for relative questions, but most that I found were people asking about invite wording or about what to do in a specific case of people wanting to bring babies. My questions are a little different.
I know not to put adults only on the invites! We will simply address the invites only to adults if we choose no children. 
Here are some of the options we've been weighing out so I'm typing them out to sort of get my mind straight and hear some opinions!
-We are getting an insanely good deal on our alcohol. My mom is a chef and dad is a bartender and their beer guy is hooking us up! Our wedding will be like a picnic and we could (very tastefully) have the beer in something like a large cooler (there will be a LOT of beer) and we'll also have wine and a few (probably around 3-4) other drinks, which we COULD just have in those large dispensers. But the point of this point -_- is that it would be nice to not have to have someone staring at these beers and drinks, making sure kids don't sneak some (there's some teenagers on his side) and there will be no question about anyone being legally able to drink (no ID checking or what have you).
-Oh God I can already see this getting long...I'm sorry ladies!
-FI's dad (their family is travelling about 4 hours to come) thinks kids should attend and have a babysitter at the hotel for the reception time so parents don't need to leave kids at home so far away. This could work I guess, but in that case, who babysits? I'm not going to ask MY friends and family to leave to watch their kids. I was thinking the older teens in his family could, but then...what do I pay them? For whoever does it-Would I have to pay them or each parent? Or not worry about it if a teen in his fam does it since they're his family?
-It just overall seems easier to not have kids there. I just wanted some suggestions as to what to do about babysitting those kids if we do choose that option.
If that was a confusing mess, I'm sorry! 
Re: Children at the wedding?
It's also rude to invite kids to the ceremony only and not to the reception.
It is a perfectly valid option not to invite kids at all.
We hired a couple people to keep and eye on the drinks, Sangria, wine and the beer are what we served other than non alcoholic. We had them pour the wine and keep the sangria dispenser filled. Beer was out in a large metal tub full of ice for people to just walk up and pull out the can or bottle they wanted. We had the non alcoholic drinks approx 20 ft way and they kept the coffee going fresh as well as hot water for the tea drinkers. We paid them each $100 bucks but it was worth it to us to make sure the teenagers didn't get into the alcohol.
There are lots of kids in my family and among their friends, and many of the other adults indulge them to the point that everything else gets ignored. I would like some time with my relatives where the kids are not behaving like this and the adults are not enabling it. I don't think that's too much to ask for my wedding, which is a once-in-a-lifetime occasion and is not going to happen again in my lifetime, unlike my birthday which will recur again every year.
"Love is the one thing we're capable of perceiving that transcends time and space."
Consider hiring a local bartender to serve the beverages to prevent under age kids from getting into booze. Or if you are doing it all diy family cooking/catering here is an idea. I don't know if you have any in your area, but I work part time for a company that just provides servers. I come into a person's event, help them with set up, then during the event go around and clean up plates, empty trash, etc, basically what a party hostess would need to do for clean up during the party. This then allows the party hostess/host to enjoy their event with their guests. When the event is over there is a fraction if any work for the hostess to do to clean up. If you know anyone that is a waiter/waitress, that won't be a guest at your event, consider offering to hire them to do this. I wear black pants & white shirt so i look professional.