Well, Rasputin probably would not have been in the picture. But Russia still had a lot of serious issues, so unless Nicholas II suddenly comes to his senses and agrees to major reform, you would most likely still have the Russian Revolution.
It is too simplistic to blame the revoluition on Alexei's haemophilia - however his haemophilia had an undoubted effect on the behaviour of his parents and assisted in making the regime more remote (as Alexandra isolated her family), however Alexandra already has an affinity for the more mystical elements of orthodoxy and both her and her husband had taken refuge in an idea of the loyal "russian peasant" as opposed to the aristocracy and the growing industrialised urban peasant.
Given the date of his birth and the behaviour of both Nicholas and Alexandra to that point, and the behaviour of Alexander III before then a healthy male heir might have assisted - assuming that their actions and the course of the first world war aren't altered too much = then Nicholas abdicates in favour of his teenaged son in 1917 who is easier for the opposition to unite around under a pragmatic regency under Grand Duke Michael which might assist Lvov's Government to survive and defeat the threat posed by the bolsheviks.
At best Russia becomes a constitutional monarchy which survives the war at worst Russia becomes a republic.