If you want to take your baby with you when you go to see ruins, Cozumel is probably not the place to do it. We've traveled a lot with a baby (both recently with DD, and years ago when DS was young), so we've got a fair amount of personal experience. What other ports are you going to?
As others have mentioned, there is a long, often rough, boat ride to and from the mainland, plus a long bus ride to get to any ruins you would see. (We've done Tulum, as well as a couple of lesser known sites.) Yes, Tulum is gorgeous and worth seeing, but don't inflict the journey to it on a baby. Even in February it might get hot, although that is generally a pretty good time of year. Getting there (ferry ride, hike to buses, close to an hour and a half bus ride, tram or hike to site, then reverse to get back to the ship) is pretty strenuous even for adults. Just because they'll allow you to bring a baby doesn't mean it is easy or pleasant to do.
Some years ago, we were signed up to do the ruins excursion on Cozumel that disneyholic did. They take you to San Gervasio. DS was really cranky that morning, so I sent DH on to the excursion and stayed with DS on the ship. DH reported the same issues with the mosquitos, and said that it was quite a hike, and he was so so glad that he hadn't brought DS. It was a very fast paced hike, and while DH kept up, he didn't think he could have while wearing DS.
If you're really wanting to see the most impressive ruins around Cozumel, that is definitely Tulum. But consider leaving your baby in the nursery for the day, if you can bear it. Or modify your plans to include the baby in something less strenuous.
Have you considered going to Chankanaab, and maybe doing the dolphin discovery interaction? It isn't ruins, but it is a national park and it is a great spot for kids in Cozumel. Even if you don't do a dolphin interaction, it is beautiful, interesting, and fun, with gardens, iguanas, a sea lion show, a shallow sort of tidal pool/cove where you can see a number of fish even if you aren't a good snorkeler, a good snorkeling beach, and a pretty good restaurant. This is pretty much the only place in Cozumel we've found that we're really happy bringing our toddler for the day. (She's been once before, and we're taking her again next month.)
Anyway, I totally get wanting your 6 and 9 year olds to see ruins. We've been dragging our now 7 year old to ruins and historic sites for several years now. We also have already taken our DD, who is just 1. (DS climbed a Mayan Temple/Pyramid in Belize on our last cruise, and since DH was wearing DD in a baby bjorn, I had to climb the temple with him. I hurt for days afterwards!) DD was walking when we did our ruins trip, so we let her walk a little, and ride in the bjorn for much of the trip. (Ours was in February, too.) It was much harder with her along, but we managed, and I'm not sorry we did.
The ruins excursion in Belize was a little easier than Tulum would be, but not too much. We had to tender into port, then wait for them to divide us into our groups, then walk to the boats that would take us up a river, take a river cruise, have lunch, bus to the ruins, do the ruins, take a bus back from the ruins, and then take a tender back to the ship. 8+ hour excursion, and we managed it with a baby. She was worn out after walking around the hot ruins, and slept the bus ride back. We were grateful. Everyone around us was nervous when they saw we brought a baby, and we were a little tense worrying that she would get cranky, but she turned out to be great and we all had a good time. The real difference between this excursion and the one out of Cozumel is the ferry ride to the mainland. It can be a real nightmare. We've done that ferry ride several times, and it's been miserable pretty much every time except once. (The last time was so bad that we said we'd never do it again, and we haven't.)
We are doing ruins again next month, but we are leaving DD in the nursery. We'll be cruising with
DCL this time, and we're so glad we have the option to leave her. (We did not have that option on our last cruise.) Yes, she was good last time, but we can't count on it, and from everything I've heard, she'll have a much better time in the nursery than she would out with us. (If she turns out to hate the nursery, we'll re-evaluate, but that's our current plan.) Obviously, what works for us might not work for you, but that's what we've decided to do.
Hope some of this is useful!
This is the first cruise we have ever taken and really want our 6 and 10 year olds to see some ruins instead of just hanging out on the beach in Cozumel. But we will also need to accomodate our 9 month old. I understand if we cannot have a stroller, but are there any excursions that will allow us to take him? 
