The optics of the situation were bad either way. As history shows, the decision to hide Alexei's hemophilia looked bad: it gave Rasputin undue prominence and made it look as though the family were completely under the thumb of a wild priest. However, Nikolai and Alexandra felt that the other option was no good either. Royals today are allowed to be somewhat more private and, not being absolute monarchs, have their personal situations and their physical persons separated from issues of state. In Imperial Russia, this was not the case.

The etiquette of the Russian court held that the health of the tsar and tsarina should not be discussed - it was essentially a polite fiction that they were always in tip-top shape. Their well-being represented the well-being of the state; if they were frequently sick, it implied that they were not taking care of the country and that the country itself was sick, in a way. Alexandra actually was chronically ill herself, particularly after Alexei's birth, and spent a lot of time convalescing in bed, which was one of the factors in the Romanovs' seclusion and separation from the kind of public life other monarchs used to control their images.

Technically, I'm not aware of a taboo on discussing the tsarevitch's health, but in an atmosphere with this kind of prohibition and where he was going to become the tsar one day, it made sense to keep it under wraps. That his problem was hemophilia added another layer to it, because it's uncurable and debilitating. Whenever Alexei was unable to attend a function he was supposed to, his parents would say that he'd sprained his ankle or had a cold or something mundane like that, because that was much more acceptable. They would mean staying quiet for a few days and then feeling better, but hemophiliacs of the time rarely survived past childhood. Now, the throne wouldn't have been in jeopardy if Alexei had died - there were many male Romanov cousins in line for it after him - but PR-wise, it would have looked very bad. There was still a kind of idea that God's favor would be shown through the transmission of a monarchy from father to son, so for that to fail would have made it seem like Nikolai and Alexandra were being scorned by God. Which might imply that maybe God was giving up on Russia as a whole, which obviously wouldn't do a lot for the people's faith in the royal family and the government.

In retrospect, we can see that going public with Alexei's condition would probably have done a lot to make it clear why the family allowed Rasputin so much license and why Alexandra was always unhappy and secluded (the popular assumption was that she hated Russia). But at the time, and with their perspective clouded by being so separated from the public, this wasn't clear to the tsar and tsarina.

I would really suggest reading Robert K. Massie's Nicholas and Alexandra: The Classic Account of the Fall of the Romanov Dynasty, if you're into this subject - it's very good pop history, and he's a good writer.