Well, my experience and thoughts differ quite a bit here, so I will outline them. Our oldest has been on a boat since he was 5 days old. We lived aboard in Fort Myers, Florida. We had 2 dogs on board too (Minature Schnauzers). My kids are now 5 & 2, so I will tell you what worked and did not work for us:
1) The cockpit was off limits unless an adult was up there. We rigged it so it was difficult to get out the doors. At anchor, we are lax about wearing a life jacket in the cockpit as long as adults are there. My youngest (2 yrs old) cannot go forward without LJ, no matter what. Oldest can with supervision. At night, offshore, storms, and anchoring/going in slip, kids wear life jackets or stay down below. The last two are becuase it takes considerable attention from me and the wife and you cannot give two things your total attetnion... but that is up to you.
2) Dogs. I love my dogs. They are great. However, I am negative on dogs on boats. They are more trouble than they are worth. It is like having two more kids. Some countries are very keen on you leaving your pets at home too (look back at other threads). I personally would NEVER consider a dog as even another means to watch my boys. I will tell you the truth: My dogs have gone into the water countless times (which is why Scooter wears a jacket too). Scooters jacket never comes off (and he prefers it that way). Think about what happens when your dog falls in the water. They CANNOT get back on a boat, period. No jacket, goodbye Fido. Also, dogs have to poop and pee. Now, let me ask you this practical question that I did not realize until we livedaboard with kids: Who is going to watch your kids when you take them out to pee (which can be quite a walk if you live in a marina)??? Take them with you? Yeah, right. As our Rhode Island friend (young female liveaboard with kids too) said, "Fooogeetabout it!" WHat about when it is raining? Freezing cold? No, no dogs. You will have enough stress without them.
3) Dogs take up a bunch of room. If you have not lived on a boat before, especially with kids, let me tell you things are going to get real tight. It is not like living aboard by yourself/spouse. Thorw a dog into the mix and things get even tighter.
4) Boarding the boat/docks. That is the only time my kids have fallen in the water. That is the most dangerous, in my opinion. Imagine being their size and trying to board the boat. Now, add in a bag of groceries, two dogs pulling at you, it is raining, the boarding ladder is slippery, you have oil on your shoes from the yard, the boat is rocking... you get the picture. Either leave the groceries in the car and get kids on first or other way around. Just a warning.
5) Life Jackets. My kids are very comfortable in the jackets... and they have no choice. Sorry. That is just life. Your kids will get very used to them. I suggest a Mustang with the crotch strap, zipper, and head float. They are the best ones made for young kids. Once they get older, say 7 or 8 or more, consider an autoinflate (made for kids). Until then, they will be fine. They will sceram a bit for a while, but then you will have to pull them out of them.
6) Life aboard. I think someone made a comment earlier about the dangers of living aboard with kids and their dreams versus yours... or something like that. Don't let someone elses insecurities scare you off. Go for it. If you live your dreams you will be a different parent to your kids. Your dreams will likely change into their dreams too. It will change your life and their life for the better, I promise. There are so many beautiful things to see that you can NEVER appreciate until you live aboard and take your kids with you. I may write a book one day on some of the soul changing experiences we have had, and will have again soon. It is not always pretty and it is not easy... but the best things in life never are. Get out of the city. Walk the docks in the evening. Watch sunsets with your kids from the cockpit. DInk around the other boats in the marina and dream. Anchor off an island and stare out into the sea. The best (emphasize BEST) people I have ever met have lived aboard around us.
7) Some cautions. Just a few words of warning, and anyone who reads this, please do not take any of my comments personally. First, your age versus the age of those around you will be very different. This is not all bad and many of our friends are older than my parents, but better be prepared for a generation gap. None of them will have kids either. It is tough to find other cruisers with kids. Few people have the money, even fewer that have the money will do what you are about to. That will give you some frustrations. Really scout out your marinas. Try and find one close to a park or that has a lot of "grass" room to run around, play blocks, ride bikes, etc. The marina will make or break you. Make a LOT of breaks to the parks or outside activities. If your spouse is not going to work (like mine did not) she will need a car or close proximity to a lot of things to do to get her off the boat with the kids... unless you want her to kill you when you get home.
8) Make it work. I promise you that the first few weeks will be heaven, then reality sets in. Make it work by getting off the boat, taking dink rides, getting out of the docks with the cataman... in other words, a lot of activities.
If you have any questions, PM me. I will be happy to help in any way I can. I am always anxious to see other parents out there trying to make cruising with kids work. And as your wife reads this, tell her, "Yes, it is worth it." It has been for us and has made all the difference since.