The incubation period for Smallpox is about 12 days. Columbus' first journey took 29 days. So this gives the travelers time to develop the disease mid-voyage. Did they truly not know they were sick and develop symptoms on the ship? If so, were there any measures taken to reduce exposure to others? Or were these men asymptomatic due to surviving the disease for generations and passing along traits that kept the virus contagious but otherwise dormant in their bodies?
I ask because astronauts go through intense physicals etc. It would seem odd that anyone would want to board a ship with a sick person, especially since diseases back in the 15th century were probably nothing to laugh about. Sharing a boat with a sick person for a month would probably not be ideal.
As colonization started before the development of the germ theory, Europeans had no direct proof they were responsible for the spread of infectious organisms among the Native American population. That said, Europeans realized infections could spread, and mortality from infectious diseases seemed to follow the Europeans as they arrived in the New World.
Europeans knew diseases could be transmitted in some way. The practice of quarantine was established in the 14th century in an effort to restrict the spread of the bubonic plague. When faced with high Native American mortality many colonists were apt to interpret Native American deaths as a sign of divine favor for European efforts. The idea of disease transmission varied among Native American populations. Some did not know disease could spread person to person, while others, for whatever reason, took protective measures. For example, by 1523 Abenaki living along the coast of Maine refused to conduct face-to-face trade with Verrazzano, preferring instead to pass trade goods via a rope over the open water. We don't know if this was a response to prevent disease spread, or an attempt to prevent Europeans from kidnapping and selling them into slavery.
We have no evidence that anyone on Columbus's first voyage was infected with smallpox. The virus arrived later, reaching Hispaniola by 1509. The first recorded introduction of smallpox to the New World mainland occurred with Cortez's 1529 venture into Mexico. It is important to remember that smallpox was only one of a cocktail of infectious organisms introduced to the New World. Measles, typhoid, influenza, cholera, and others added to the infectious disease mortality.
As an aside, pathogens constantly evolve and modify their virulence in response to changing host/environmental conditions. There is no reason to assume all New World epidemics were the result of introduced organisms. For example, a cocolitzli epidemic that burned through Mexico in 1545 and again in 1576 was likely caused by a Hanta Virus-like pathogen native to the New World. When the environment changed, like the extreme drought conditions seen those two years, the opportunity for pathogen transmission to human hosts changed and resulted in an extremely virulent epidemic with high mortality.
There is actually much scholarly debate on the topic. This is a famous example from later colonial history. This is from a scholarly article discussing this very topic, and one of the few cases of evidence we have for it.
During the Indian resistance to British imperialism in the Great Lakes area (Pontiac's Rebellion, 1763-1764), Amherst brought up the idea of germ warfare in writing to Colonel Henry Bouquet (their correspondence is preserved in the British Museum). Scholars dispute the handwriting, signatures, chronology, authenticity, responsibility, and outcome. According to some, Amherst was only recommending biological warfare when he suggested in a letter sometime in 1763, "Could it not be contrived to send the small pox among the disaffected tribes of Indians? We must on this occasion use every strategem [sic] in our power to reduce them." Bouquet wrote back in July, "I will try to inoculate [them] with some blankets that may fall in their hands, and take care not to get the disease myself" (Heagerty 1928:43; Hopkins 1983:246; McConnell 1992:194; Simpson 1980:30).
Others stress that Amherst issued a direct command: "Infect the Indians with sheets upon which smallpox patients have been lying, or by any other means which may exterminate this accursed race" (Utley and Washburn 1977:98; Wright 1992:136-137). Still others quote Amherst thus: "You do well to try to inoculate the Indians by means of blankets, as well as try every other method that can serve to extirpate this excrable race" *(Knollenberg 1954:492-493; Parkman 1991:648). *
The Nessus Shirt in the New World: Smallpox Blankets in History and Legend Adrienne Mayor The Journal of American Folklore , Vol. 108, No. 427 (Winter, 1995) , pp. 54-77
Or were these men asymptomatic due to surviving the disease for generations and passing along traits that kept the virus contagious but otherwise dormant in their bodies?
No, smallpox is only contagious when you have an active infection. Once the lesions go away, you are not contagious. However, dried lesions on clothing or bedding can infect others.
See p.19
Did they truly not know they were sick and develop symptoms on the ship?
You are assuming that the spreading/killing rate of smallpox among Europeans was the same as in Amerindians. Obviously, it wasn't. Most Europeans old enough to join a crew and sail the Atlantic would have already been exposed to the disease and survived it (it was most common and deadly among European children and teenagers back then). Most people aboard this given ship, in consequence, would be immune to smallpox and not get the infection from the one or two odd adults in the ship that was suffering from the disease. These sick Europeans wouldn't be as likely to die as the natives either. What caught the Europeans' attention was not that the natives were catching smallpox and other diseases, it was that so many of them caught the diseases at the same time, and so many died of it so quickly, rather than recovering. That was not how things usually went in Europe.
Now, as anthropology_nerd said, it wasn't Columbus who introduced smallpox in the New World anyway and it arrived later to the Caribbean. The great plague wave that hit the Aztecs and also caused the Inca Civil War in the late 1520s is believed to have arrived from Cuba in an African slave that was part of the Narvaez expedition against Cortés in 1520. But that doesn't mean this African slave brought the smallpox and was suffering from it all the way from Africa. He could have caught it in the Havana docks the day before leaving, for all we know.
I'll do my best to answer this, but my memory is sketchy:
As Smallpox was pretty prevelant in the 15th and 16th Century Europe, most of the crew that Columbus took with him on his voyages would have already have come into contact with the disease and, most likely, would have already survived the disease, or were naturally immune. However, this does not mean that they could still carry the pathogens.
It is possible that some members of the crew contracted smallpox on the voyage. but there is no way of knowing for certain and this is pure speculation on my part.
What is most likely, is that the crew would have carried the smallpox pathogens with them from Europe and taken them into the New World without realising it, where it would have devastated the population that had never come into contact with the disease before.
This site makes the claim that an infected African slave in 1520 brought the disease over to the Americas.
One of the first known instances of biological warfare was during the Siege of Fort Pitt during the Pontiac's War (1763-63) when the British gave blankets infected with Smallpox to the Lenape (Delaware Indians) (Anderson, Crucible of War). Where there instances before that? Possibly, but I do not know.
Tl;dr At first, probably not. Later, it was used as a weapon.