Hi Margaret
One of the dilemmas that we have as hosts, which many hosts may not fully appreciate, is that when we are offering accomodations in the US (and elsewhere) we are limited in what we can say in our advertising by the Federal Fair Housing Act. The Fair Housing Act states that:
"The Fair Housing Act protects people from discrimination when they are renting, buying, or securing financing for any housing. The prohibitions specifically cover discrimination because of race, color, national origin, religion, sex, disability and the presence of children."
What most people don't understand about the Fair Housing Act, is that even though most people who are renting out a room in the home they live in are exempt from this law in the sense that they are permitted to discriminate on any basis, in terms of their selection of renters, the law still applies to everyone in that it does not allow discriminatory STATEMENTS in any advertising. THis meaning and this aspect of the law is not really captured in Airbnb's Help Page on the subject, but if you look at Craigslist FAQ on the subject you will see the info presented clearly there.
Hence, in order not to run afoul of the FHA requirements, hosts should avoid stating in their listings that they don't permit guests to bring children. Your state may also have laws against age discrimination which may mean you should not state in your listing that you don't accept children (or adults!) under such and such an age.
As far as I understand the law, hosts can privately decide not to rent to children, or to not rent to anyone under a certain age, but just can't state this in their advertising. Which is disappointing and frustrating for many, as doubtless this leads to wasted time and energy when you get inquiries from those who are not a good fit for your place. But, so many in the USA are litigious, and there are some who probably make their living trolling through housing ads trying to find people whom they can sue, based on a few words that apparently seriously injured them (rolling eyes). Hence it pays to be cautious.