The bulkhead issue was in the early 382s. They were repaired and the 383/384 have no such issues. Neither does my 382 #163. The aft head bulkhead on my boat was properly tabbed to the hull. The airex core goes to the waterline; solid glass below that. The boats are not massively built and I would hate to run one hard into a coral reef, but few boats not built of steel or aluminum will withstand hard groundings without significant damage. But Morgan's are better than Hunters or Beneteaus, in my view, with their internal glass liners and bolt on keels. At least one Beneteau lost its keel in mid-ocean. The issue is, each Morgan is different, despite the same design and manufacturer. Depends on who was working on the boat and how the work was scheduled, I think. For instance, I have used my boat pretty hard and all my bulkhead tabbing is solid. Another owner, who sailed both ways across the Atlantic had some bulkhead tabbing give way--a bad situation. For what you have to pay, they are generally a good value, but there are other boats I might buy if I were going offshore and I had somewhat more money. And as Mark says, it is often about upkeep and maintenance. I bought my Adavida 23 years ago for $57K for inland cruising, and then went offshore. Before I went seriously offshore, I cruised up and down the Washington Coast is some pretty nasty conditions,
Then, before we took our 10 month tropical journeym I replaced all standing rigging, installed offshore equipment like AIS, water maker, Monitor windvane, Satphone, storm jib and main trysail. She is probably worth less now than when I bought her, although I got a surveyor to value her at $55K a couple of years ago and she now has a fresh coat of paint. But no one owns a boat as an investment. It is about the love of sailing.
FWIW, I hit a reef in Fiji at over 5 kts. Brought the boat to a dead stop, and the bow dipped into the water. Scared the hell out of me, but only cosmetic damage on the leading edge of the keel. If it hit the hull it would not have survived. If anyone is venturing that way, the charts (even "up to date Navionics charts" ) are wrong, and so are the waypoints you can get locally that have been sailed "hundreds of times."
The hull is solid below the waterline, airex above, and the deck is plywood core. The plywood is cut into squares, so if water gets in there there is a break and it won't spread.
The boat can be uncomfortable downwind, especially if there is a swell on the beam. The Indian ocean sucked, but many will say that is true regardless of what monohull you sail there. The south Atlantic crossing was a pure dream, with 2 weeks of not touching sail trim or steering trim. The Pacific was hit or miss, but mostly comfortable, other than the many gales I hit due to my late start (Left Hawaii in August).
The cockpit mounted traveler is a serious hazard on the ocean. Some like it for single handing, but as a single hander I'm very scared of it. Accidental jibes happen, and I have had my GPS ripped off the pedestal by the main sheet, had the sheet grab gear and throw it over(lost both a radio and a winch handle), and once had it catch me and slam me into the cockpit combing. So if I did it again I would get a later model with the traveler on the deck. If I can get it in the budget, I will add a staysail stay before I head out again. The designer (Ted Brewer) drew plans for this, and he is still with us and those places are readily available. It is a straight forward upgrade. Also, the 382 doesn't have dorades. It gets miserable on hot humid days in the rain. So again I'd go for the later model.
Work with your sailmaker on sheeting angles for the genoa. Let him know you will be in the ocean, and fulling it very small. I run out of track and so as I furl the top twists off and I lose the ability to point. It's fine downwind, but in 30 kts, I can't sail higher than a beam reach. A properly sized staysail would negate the need for this compromise on the genoa shape, which is why I am looking at that upgrade.
The rigging is pretty stout. I see many boats of similar displacement with 1/4" stays sailing in the Ocean. Also, the chain plates and turnbuckles are sized for 3/8". The turnbuckles on my Morgan are much larger than those on other boats with 5/16" wire.
I have some serious steering repairs to make after 30,000 miles. But, 40 year old parts made it 30,000 miles. Much of this damage is the result of opting for a monitor windvane over an autopilot or hydrovane. So you could probably avoid it.
Overall, the question is going to be the condition of the boat you are looking at, and your budget for the boat and refit. All of the standard stuff for fitting a boat for the ocean applies.