It would have changed a great deal, because Alexei's condition greatly impacted Alexandra and she and her outlook greatly impacted Nicholas II. Nicky was that rare breed of men who are stubborn and people-pleasing, which meant he had his own views on a great deal of things and many of them awful and dumb and Empire-wrecking, but it also meant he did not disagree with his advisors even when he was given advice he did not agree with because he did not wish to seem argumentative and wanted folks to think he had listened to them. Then, he merely waited for an advisor to pipe up and voice the opinion he actually did agree with and "go along" with it. Naturally, this made the advisors with whom he pretend to agree with feel left at the altar and think him weak - a cushion which bore the imprint of the last ass it came into contact with, and this colors our understanding of him to this day. So I do not think Nicky was so weak as to be manipulated by his wife and her mesmeric charlatan, but for the sake of family, children and keeping the peace in the house he did humor her beliefs, at it was to the detriment of the monarchy.
What doomed the Russian Empire was World War One. What doomed the Romanov dynasty was how Nicky handled the War. Alexei being a healthy baby boy would not prevent the Great War, but it sure would have modified how Nicky reacted to the War and how his wife reacted to various changes in government and war developments. I'm not going to play pop-psychologist, but other historians have, with regards to Alexandra and her motivations. One interesting case study on Alexandra was written by a historian who himself had a child with hemophilia and he talks, with depth and expertise, on how parents of hemophiliacs react and overreact. The theory holds by finding herself dealing with a condition she had no control over, she tried to exert control over things which she felt she could and aught control to balance the scales. Would a less morose Alexandra with a healthy heir to secure her husband's dynasty be as apt to get involved in the Russian government? Maybe, maybe not. But it certainly would have helped.
Would Nicky, with a happier wife and a secure future, make the exact same blunders he committed? Probably not. But he would have committed a slew of new ones instead. He was an Edwardian figure thrust into ruling an allegedly modern state with a feudal and archaic infrastructure and fight a new sort of war. Many much smarter men than him spent three years killing their soldiers on the Western Front before learning how war is waged. And here, he would have been asked with not just understanding how to fight a War, but also rule a state in a new way - which would have contradicted his upbringing.
My thinking: the Russian Empire is doomed and so are the Romanovs, but maybe it doesn't all end in a basement murder.