I had actually never heard about Baldwin IV before your question (though I found a very nice article about the cultural and medical intersections in Hektoen, below), but I can comment a little of historical views of leprosy. Leprosy (Hansen's disease) is called by mycobacterium leprae (the same genus that causes tuberculosis), and is notoriously difficult to transmit. It's spread through respiratory droplets, and even in close contacts, less than 5% of people will contract the disease. It's treated with a long course of rifampin and dapsone (+/- clofazamine). Unlike TB, there hasn't been a large amount of resistance noted, and success rates are incredibly high. There are basically no cases in high income countries anymore (less than 200 per year, and almost all in travelers).

That's not to say that leprosy isn't a problem -- it still is in LMIC, and is a disease of poverty. But where does the view of leprosy as a scarily infectious disease from? If transmission rates are that low now, what happened? Has the disease become less virulent? Is incidence just down so much that relative transmission is well? Probably both these things... but probably the bigger reason is that nosology, the classification of diseases, did not really exist yet, and diseases were largely described by their symptoms. And it just so happens that a ton of dramatically different skin diseases can, untreated, present like leprosy. You have nutritional deficiencies (especially pellagra, which was called the "Asturian leprosy" in its early days), fungal infections (google image "favus" if you've got a strong stomach, and be thankful for topical antifungals), plaque psoriasis (again, you are forewarned before clicking that picture, another reminder of how amazing simple over the counter steroids are), cutaneous tuberculosis and scrofula, let alone the myriad types of skin cancer... the list goes on. Early leper colonies, where these people were isolated, were probably filled mostly with people who did not have leprosy.

Leprosy is, in fact, an old disease, and probably appears in the Hippocratic corpus. We also have DNA analysis showing its ancient origins in Asia Minor. But we should be skeptical of descriptions of descriptions of leprosy before the early modern era. That being said, from what it appears, Baldwin likely did have the disease -- initially with nerve involvement -- the archbishop William of Tyre noted: "Half his arm and right hand were dead … he could not feel if pinched or even bitten.” Later on in his life, he developed skin ulcers, facial deformities, and stereotyped deformities of his hands and feet, and finally blindness. The first sign of the disease is usually depigmented painless skin patches, followed by increasing deformity of the hands, feet, and face in later life. And according to the Hektoen article, no one in his household was reported infected with leprosy -- which is actually incredibly consistent with this difficult-to-contract disease.

So, the TL;DR of your question -- 1) he was probably the unlucky 5% who got it from a close contact and 2) he didn't infect his household because he likely actually had an m. leprosae infectious (as opposed to something like a fungal infection) which is quite difficult to spread.

As for the Hektoen article I used: http://hekint.org/2017/01/30/the-remarkable-baldwin-iv-leper-and-king-of-jerusalem/

Hektoen is one of my favorite medical journals -- all about medical history and humanities AND committed to being open access! Highly recommend it!

EDIT: Sorry for the gross pictures/google searches ...