Why?
Last I checked, Seung-Hui Cho isn’t on the Enterprise.
The risk that Romulan warbirds present to a teenager on the Enterprise isn’t affected at all by whether or not his mother is on the ship.
So you'd be perfectly fine leaving your 15 year old kid alone with a few people you've just served with on a military base under threat of attack while you're posted back in the States just because he enjoys the experience there? That's pretty much what this is.
If it weren’t a safe place, I don’t think I’d have moved my kid there in the first place. You’re arguing against families being on the Enterprise in the first place, not against allowing them to stay when the parents leave. The people who actually live in the universe under discussion consider the Enterprise to be a safe enough environment for children.
And it’s not “just because he enjoys” the experience.” Wesley’s interest in serving on the Enterprise isn’t hedonistic.
As for me, if my son were 18 and qualified to serve, I would certainly allow him to do so. I might or might not be comfortable with it, but if serving in the military or Starfleet is what he wants to do with his life, of course I would allow him to pursue it. Wesley is about 17 years old at this point—Memory Alpha places his birth in 2348 and S2 in 2365—and physical durability isn’t as much of a requirement on the Ent-D as it is in present-day armed forces, so the situation is basically the same.