“How can we nicely tell our guests that their children are not invited?”
In general, weddings are grown-up events and usually run well past bed times. The basic rule is that the names of the family members that are invited are the only ones on the envelope. If children are not invited, their names are left off the envelope.
Of course, most people don’t know this point of etiquette and a line on the invitation that says “no children, please” seems mean. The best advice is to talk with your family and friends with children and let them know your wedding is for adults only. This is good job for your mother or your bridal party. You could also put this bit of information on your wedding website.
If you think it absolutely has to go into the invitation package, I recommend a line at the bottom of the invitation that says “Adults-only reception to follow.” This can also be handled with a notation on the reply card, where people have a better chance of actually reading it. A delicate note of “Adults only, please” should handle it. If your ceremony and reception are at different places, you’ll be using a reception card. It, too, can say something like “Our celebration is limited to adult guests.”
There was a time when weddings were considered adults-only affairs to begin with. Children, such as flower girls and ring bearers, were shuttled off after the ceremony not to be seen or heard from again that day. There will still be guests who insist that their children are not a problem; but that is not really the question. It is the perogative of the hosts who is invited and who is not.