I would be inclined to think that alaska would have become something of a penal colony. A place for the czar to dump political dissidents. Maybe they would have set up a naval base there, with some strategic importance in the Pacific.
Also, If Russia had discovered the Alaskan oil, then the Russo-Japanese war, and the IWW (if it happened at all) would have gone completely different. Imagine if Russia had tapped the oil wealth of Alaska! they may have been able to stand up to the Germans!
A post-IWW surviving Russian Empire would have had enormous butterflies.
But, assuming that didn't happen...
The Russo-Japanese war may have been carried over to Alaska, and this closeness may even have prompted British/American intervention. The Americans may have itched to sieze the territory under the casus belli of the monroe doctrine (that would have changed the conditions for the IWW drastically).
Also, if all those communists and republicans I mentioned before had been sent there, then there may have developed a strong, closely knit community of dissidents. They may (if the IWW had gone roughly the OTL course) have tried to set up their own state in Alaska, or they may have taken ships from the naval base and made their triumphant return to Russia, crossing through Siberia and inciting rebellion.
Then the US or Britain would have siezed Alaska, and if the struggle was prolonged then the Germans may have gotten the upper hand.
After the Russian civil war, the Soviets (and possibly the Germans) would have demanded a return of Russian Alaska. The Soviet Union would have come into conflict with the Germans, and may have allied with a revanchist France (possibly under maurras) in a very different IIWW!
very interesting butterflies.