How to Indicate an Adult-only Reception?

It's completely legitimate to want an adults-only reception, especially for an evening affair. And most parents of young children will jump at the chance for a night out without the kids. Even so, this is a sensitive topic, and putting "adult reception" on invites is not the answer. It seems like the easiest way to deal, but it's a little too in-your-face, so you should take a more subtle tack. First, tell your parents, wedding party, and other close relatives and friends, so they can spread the word if any guests ask them.
Second, the people whose names are on your invitations are the only people invited to the wedding (i.e., "Mr. and Mrs. John Doe" means just the couple; "The Doe Family" means little Suzie can come too). Most guests will take note of this and RSVP for just themselves. Others are not so observant and will RSVP that three guests will attend, even though only two were invited. This is how you'll know if they think children (or random other people, for that matter) can come.
The next step is to call them and explain that because of "budget constraints" (always the best excuse, even if it's not true) you decided to invite only adults. If you meet with anger or exasperation -- and you might -- don't back down. It's your decision as to who's invited to the wedding. Whatever you do, don't make exceptions. Don't let Suzie tag along just to get off the phone with Jane Doe -- otherwise little Johnny's parents will notice.
This seems hard, but it's only awkward because many guests do not gracefully accept the fact that their kids aren't invited. Try to understand that some may be genuinely surprised or hurt, and be understanding, but don't give in. If it becomes a real problem with a lot of guests, look into hiring a babysitter or two to care for kids during the reception. They can have a pizza party -- way cheaper than having them at the reception, and everyone will be happy.