IIRC, the reason for killing the Romanovs was to prevent the White Army from taking possession of them, as they were pretty close to it and were likely to take it as is (and they did IOTL after the execution of the Romanovs). The entire point of murdering the whole of them was to prevent a royal from surviving and being able to rally the White cause and get more help from other sympathetic powers.
Assuming Anastasia lives, she's going to be a figurehead. The Whites will use her to raise support for more foreign investment and intervention into the Russian Civil War, and we might end up with the wholescale defeat of the Russian Revolution. If that happens, she gets crowned Empress and they probably draw up some vaguely liberal constitution that renders her pretty much powerless. Then we have a really shitty time (a 'Weimar Russia', perhaps?) before a White-fascist dictatorship or something like that.