I hope Ed is right about most museum models being kept in cases and/or controlled atmospheres. I don't have any statistics, but I'm afraid he may be mistaken.
A lot of people have big misconceptions about how a maritime museum works - especially regarding ship models. I found that out when I got a job as a curator at the Mariners' Museum, way back in 1980. Believe me, there's no big staff watching out for, and working on, the models. So far as I know, nobody on the current staff of that museum has ever built one.
Three staff positions (or possible staff positions) in a well-run museum have something to do with models: the curator, the registrar, and the conservator. The curator is the "subject matter expert" of the museum. In a history museum, he/she will typically have an advanced degree in history; in an art museum the degree will be in art or art history. The registrar takes care of the artifact collection, and is responsible for making sure that the artifacts are stored in a safe, approved manner. The conservator's job is to work on the artifacts - restoring and repairing them as called for. His/her professional training typically will be in physics and organic chemistry. Only a handful of American educational institutions offer degrees in conservation. They produce a few dozen graduates per year - only about half of whom find jobs in the field.
The vast majority of museums don't have conservators on their permanent staffs. In the state of North Carolina, which has lots of museums (and where I live now), I'm aware of three museums with conservators on site: Tryon Palace State Historic Site, the North Carolina Museum of History, and the North Carolina Museum of Art. (Tryon Palace may have lost its conservator position recently, due to staggering budget cuts.) Most conservators are free-lancers, who are hired by museums (and private individuals) on a contract basis.
I'm aware of five maritime museums that actually do ship model conservation, in-house, on a regular basis: the Smithsonian, the Naval Academy Museum at Annapolis, the National Maritime Museum in London, the Nederlands Schipvaart Museum in Amsterdam, and the Prinz Hendrik Maritime Museum in Rotterdam. (I'm sure there are a few others.) I suspect the conservators at those museums are kept busy most of the time working on other types of artifacts.
I got hired at the Mariners' Museum for (I think) three reasons: I had a Ph.D. in history (with a specialty in naval history), I was willing to work for peanuts because I desperately needed a job, and I knew something about ship models. Nobody else on the museum staff did. I was the "assistant curator in charge of three-dimensional artifacts." (There were two other curators - one in charge of the full-size boat collection, and one in charge of paintings and other artwork.) Shortly after I got there, my boss (who had a Ph.D. in American Studies) taught me the basics of the theories of conservation. During the next three years (until I reached the point where I couldn't afford to live on $14,800 per year), I worked on quite a few of the museum's models, doing restoration, cleaning, and repair work. My two biggest projects involved the museum's collection of large-scale builders' steamship models and the infamous Crabtree models. I also got to work, for about an hour, on the museum's one Napoleonic POW model. (That was just about the most neurotic hour of my life.) The project I'm most proud of was a major restoration job on a 1/25 scale model of a lightship, built in 1876.
I may be mistaken, but so far as I know nobody has touched any of those models since - other than to move them from one place to another.
The MM has pretty good environmental controls in its galleries and storage areas. In my day, at least, every model on exhibit was in a glass case. The hundreds of models in storage were kept on metal shelves, generally in the dark (that's good, conservatorially speaking), with sheets of feather-light clear plastic draped over them.
Back in the 1940s through the 1970s the Smithsonian built up a huge collection of ship models, largely due to the influence of the late, great Howard I. Chapelle. He collected lots of old models, and commissioned quite a few from the likes of Donald McNarry and Robert Bruckshaw.
The first time I went to the Smithsonian, in 1967 I think, hundreds of models were on exhibit. The thinking of exhibit designers has changed since then. The Smithsonian now has a "hall of maritime enterprise," which has a few dozen fine merchant ship models in it, and a big exhibition on American military history, which has a few. That grand old 1/48 model of the Missouri, known and loved by generations of model enthusiasts, is gone now - presumably in storage. Most of the models Chapelle had collected or commissioned also were put in storage, and made available on loan to other museums. (I got Donald McNarry's superb 1/92 Constitution for the MM. I don't know whether it's still there or not; it wasn't on exhibit the last time I visited.) The good news is that the Smithsonian's storage facilities are state-of-the-art in terms of environmental controls; I'm sure those models are well cared for.
I haven't been "back stage" at many other maritime museums. I have the impression that the Navy Museum in DC has a fine collection and good exhibition techniques. (There's a Curator of the Navy who has jurisdiction over all three-dimensional artifacts and artwork owned by the Navy.) I have the impression that the various museum on the west coast and the Great Lakes take good care of their models, but I don't think they've caught up in terms of environmental controls. I hope I'm mistaken about that.
I've seen some pretty awful sights in maritime museums. In 1978 I went to the Science Museum in London, and frankly I was appalled at how that place treated its magnificent ship model collection. In 1978 there apparently were virtually no environmental controls in that building. C. Nepean Longridge's famous HMS Victory was literally coming apart at the seams. The red dye in the banners of a French galley had faded to pink, due to improper lighting. As I understand it, the Science Museum has been redone since those days; I certainly hope so.
This post is way too long. Unfortunately, a lot of museums don't give their ship models the same priority they give to paintings and sculptures. I wish that weren't the case, but I'm afraid it is.