Well, lot's of People giving their opinions, so here's my two cents.
The problem is that the question makes us debate between pragmatism and idealism. Is it right to shoot a boy? Of course not. What if the boy can be used as a tool for a civil war, and his escape is imminent? Remember the french bourbons, remember the english restoration before that. It's a dangerous affair. Do we put ourselves in the shoes of the bolsheviks? For this example I will.
Tsar Nicholas II Romanov, Autocrat of Russia : A Big Public Trial. I like trials, I believe they give a sense of public justice by allowing the people to confront the accused. The Bolsheviks, by executing the royal family quickly in the middle of the night, deprived the russian people and their own cause of a sense of righteous justice. Therefore, charges must be brought up agaisnt the Tsar. Since there's no precedent, I imagine they could be something like: "Abuse of power", and "Crimes against the Russian People". The bloody sunday massacre could be a good example of both. There can only be one veredict of course: GUILTY. And there can only be one punishment: DEATH. Firing squad might seem a little bit too partisan, showing soviets shooting the Tsar. Hanging has been the standard method of execution of the common man(and he's now a common man, as is everyone else) for centuries. Hang the man. If they really want to be dramatic, bring forth the guillotine.
Tsarina Alexandra Romanov: Trial besides her husband as a accomplice to the abuse of power and perpetuation of imperialist oppression. Veredict: GUILTY. Punishment: DEATH. Gods representatives go meet their manager together.
The Princesses: Even though all with the exception of Anastasia are adults, they were not directly involved in politics. Shooting spoiled children will prove little. Send them to different collective farms around the country(preferably far from civilization) with loyal bolshevik garrisons nearby to check on them from time to time. Change their names, make sure it stays changed.
Alexei, the crown prince: He's a 13 years old boy with a serious physical condition. Send him to Petrograd or Moscow, put him into a very confortable room, keep him guarded and watched at all times. Teach him the values of communism. Use him as a propaganda tool for the bolshevik revolution, "The prince that chose Marx". Most important of all, control his access to the outside world, including who he meets.