Putin is not dangerous because he's insane: he's dangerous because he's extremely intelligent and calculating.
The Russians would not invade Alaska: their long-term geopolitical goal is to expand to their four coasts: the Pacific (done), the Baltic (done, partially: St. Petersburg is vulnerable to being cut off), the Black Sea (done, partially, no good port in the territory they control: they'd want Crimea), and the Indian Ocean. Alaska does not fit into this goal, which I suspect is part of the reason they were willing to sell it in the first place. And unlike Americans, they think in the very, very long term.
Also, the logistics of invading Alaska are pretty mind-boggling. Petropalovsk, the main Russian base on Kamchakta, has to be resupplied by sea because the cost of sending goods over long, sparsely populated distances is too high. Not to mention the fact that if they were so crazy as to invade Alaska, they'd face a militia in every town they tried to occupy, and would constantly have to deal with the USN and USAF interdicting their supply lines.
