If Russia does hold onto it, as Ricardolindo says, it's going to end up with more non-Russians than Russians and probably be a huge headache as a result. Can you enforce Russian law on American settlers when there's a big sea in the way and America's right there?
In the long run, it'd mean the United States and Canada have the Russian Revolution and civil war pissing on its lawn. Do Russians flee to Alaska and if so, which ones and when? Is Alaska this holdout bastion of White Russia protected by the US Navy? Does Alaska go communist and then the US & Canadian army storm in or try to blockade it? Whatever happens, US and Canadian socialists and anti-communists will both get fodder from it.