How large of an increase would be within Russian capabilities? I'm uncertain whether it would be enough to contain the gold rush; the numbers disparity is rather massive, after all. Also, the British would probably not be happy about Russia establishing a large, standing force in Alaska.
Further, the Tsarist regime had its share of corruption problems, and Alaska is almost certain to be a very undesirable posting. Even if Russia can build a large enough force to keep gold rushers out, the locals might not be too keen on enforcing the Tsar's policy. Between low morale and a lack of supervision from Moscow, the soldiers in Alaska might easily be bribed into looking the other way when American prospectors show up. Then there are worse possibilities, like some of the soldiers and other officials deserting and joining the gold rush themselves...
Also, the Russians might not have time to reinforce their presence in Alaska once the gold rush starts. After all, it would probably take quite a while for word to reach the Tsar, and for any reinforcements to be organized, and then the time to ship them to Alaska. Americans are a lot closer.