First, some very sage advice given by my peers... I'd point out that many people who get seasick on powerboats do not get seasick on sailboats for some reason, and that there is a very big difference between monohull sailboats and multihull sailboats, and some are much better for some people than others... One of my good friends, who is now on her boat awaiting a weather window to cross the Gulf Stream, used to get violently seasick on a monohull sailboat....any monohull sailboat. She tried a catamaran and the issue has become much less of a problem—it hasn't gone away entirely, but it is now a reasonable thing to deal with.
Yes, a teenager can get adapted to a cruising sailboat lifestyle, provided he actually has an interest in doing so. One who gets violently seasick and doesn't care at all for sailing probably wouldn't do well....but if he loves to sail and loves spending time on a boat, then the adaptation is relatively simple.
I would highly recommend that you and your husband take at least a basic ASA 101 course—preferably separately—and get all three of you out on sailboats in varying conditions. Getting some time on sailboats is key to figuring out whether this is doable for you and your family.
Ed's points about you and your husband both being able to handle the boat single-handedly is a very key one. Cruising couples, even with a 13 year old son along, are often effectively doing just that.
If, after you get some time in sailing, and you and your family decide to make the leap.... start off with a reasonably sized boat that you can singlehand if necessary and start by day sailing and then weekending, and work your way up to longer and more challenging passages. Not only will this allow you to build experience, but it will also let you learn about your boat and its quirks.
You should also do as much of your own maintenance and repair work as possible, since these are skills that will save you money and possibly save you and your family in the future.
BTW, this is highly doable. My friend, who I mentioned above, never sailed before the last two years or so AFAIK
. Yet, this past year, she, her husband and son, have bought a boat, spent much of the spring and summer learning how to sail her, and are now on their way to the Caribbean.
Needless to say, I'm damn proud of the three of them.
I'd point out that a boat that is capable of doing this is not necessarily a huge expense. Another friend of mine recently bought a 30' catamaran that would be pretty good for a family of three to go cruising on. The boat was relatively inexpensive, and with a bit of sweat equity, some upgrades for cruising and a new set of sails, they could easily be cruising for less than $35,000 all told.