The greatest threat would be dehydration.
Depending on the tech level, there are certainly ways to procure fresh water while out to sea (I'm going to assume you're not talking about living on large freshwater lakes). Desalinization, rainwater catchment, and dew collection are all viable sources of potable water. Like with real-life live-aboards, they would use seawater for most cleaning and sanitation.
The problem is that "post-apocalyptic" and "forced off the land" imply haste. Are there enough boats for everyone to live comfortably? What kind of supplies are there? Enough water stored to last for a couple weeks at least? Is there water procurement equipment on board, or will they need to create that?
Except in very dry climates, dew collection is super low tech and easy to do. Just use a tarp, something to hang it just so, and a bucket. Times however many people you have. More info here on dew and rain catchment.
This only works if you have tarps or other collection/condensing items and buckets/bowls/etc. And storage if your collection is not daily.
Your second biggest threat is sanitation/disease/toxicity.
That lovely rainwater that fell down? Filled with God knows what. In our current world, even rain in the middle of the ocean comes down with chemical residues from air pollution, bacteria, etc. Plus whatever you pick up from your catchment system and storage. Mostly, fresh rainwater is pretty safe to drink...but in your world? Who knows?
Toilet waste is pretty easy to deal with. If the boats have toilets, just open them to the ocean (they flush with seawater). If not, people can hang out relevant parts overboard or use pots and empty them. Saltwater should be a pretty good way to clean pots and hands. But none of it is perfect and a waste-spread disease will spread. There won't be toilet paper or showers, though people may be able to bathe in the ocean or with ocean water.
Close quarters will spread diseases of all kinds.
Food preservation problems will also cause disease. You can't catch and eat everything fresh every day. There's too much humidity on a boat to dehydrate foods. Some boats have holds for live fish, most don't.
Toxicity is an issue in our current world. My live-aboard friends tell me they never swim or fish or harvest seaweed from where their boat is docked (in multiple locations too). That's where city water drains to the ocean and then there is waste from the other boats (they're not supposed to dump there but...).
I havent been able to take my kids to the beach since we've been here. Coliforms haven't been less than "dire" even once. Plus MRSA, plus flesh eating bacteria, plus brain amoebas...We get raw runoff from Houston sewage plants every time it floods,
which means every time it rains...Chemical plant "spills" happen weekly. (Quotes from friend when living in a marina well
south of Houston)
In a disaster that includes torrential rains (Harvey for example), everything gets a million times worse. And in your post-apocalyptic world, those sewage plants aren't running anymore. Even if no one is left on the land to create sewage, there's still tons of it left. And it's all pouring into the oceans. Goes out pretty far too. And is carried by currents all over the world (like plastic).
Most factories and energy suppliers are by the coast or a river that drains to the coast. Without people to staff them, a lot of crud will go into the ocean (heck, even with people to staff them).
Oil rigs in the ocean may start to spill over.
Air flow picking up residues from explosions and spills and open pits of waste will travel the oceans too.
None of this takes into account the exposures to bacteria, viruses, heavy metals, and misc toxins during the apocalypse and in the time afterwards before getting on the boats. Or the boats being in the harbor waiting to load up and leave, then navigating to then away from the immediate coast.
Other threats include:
- Food. Seafood and sea vegetables forever. Yea. But do they have the tools to catch, store, and prepare it? Is it safe from runoff/air quality? Is there a consistent supply?
- Each other. How crowded are these boats? How deranged are the passengers?
- Storms. No more GPS or radio updates.
- Lack of repair parts. Especially for sails and rigging, which is the only way to move the boats once fuel runs out, which it will quickly. You can't row huge boats.
- Lack of knowledge about the boats. They are more complicated to maintain than most people think.
- Lack of building supplies to make floating platforms or whatever.