How prevalent were crimes like murder, arson, and robbery in Alaska/Canada during the Rush?
It largely depended on whether you were on the Yukon side or American side of the border. In general, however, violent crime was limited and most crimes consisted of property crimes (theft, embezzlement, con games, etc.) or "sin crimes" such as illegal gambling, prostitution and illegal liquor manufacture/distribution. Events like the murder of "Soapy" Smith in Skagway are remembered precisely because they were uncommon.
On the Canadian side, the North-West Mounted Police ran things extraordinarily smoothly. On the American side, things were far more rough and ready due to the fact that the U.S. government had barely any presence in Alaska at that point.
Pierre Berton's books are the quintessential histories of the period, but I prefer the works by Michael Gates, largely because Gates is still living, still producing, and he's a pleasant man to eat lunch with. Jeff Brady's Skagway: City of the New Century is awesome if you're looking into things on the American side of the border. I'll also recommend Lael Morgan's Good Time Girls of the Alaska-Yukon Gold Rush, which is THE book to read if you're interested in prostitution during the gold rush period. If you're into first-person accounts, you can pick up Sam Steele's Forty Years in Canada, which is in the public domain. (But I wouldn't trust it all that much, since it's a memoir, and he doesn't include a lot of the juicy stuff.)
Anyhow, let's turn back the clock to 1896.
In that year, Circle, Alaska laid claim to the title of "World's largest log city," as more than 1,000 people lived in the town along the Yukon River. Gold recovery in the Fortymile region along the border between Alaska and Canada was down, and it seemed like Alaska was collecting all the prospectors. There were already plenty of them, (albeit numbered in hundreds rather than the thousands and tens of thousands who would come later), and they moved from spot to spot as claims popped up.
The big gold-producing area was Juneau, in the Alaska panhandle, but Juneau was under the control of the industrial mining interests, and prospectors who thought they could make it on their own headed north instead. They traveled over the Chilkoot Trail (already well-established) and up the Yukon River on steamboats (of which only a handful were operating).
The NWMP operated out of Fort Constantine, not far from the (now defunct) town of Fortymile on the Canadian side of the border. Business was slow. The post had opened in 1895 but only heard one case every two months or so. The first of these, in August 1895, was an interesting one in which a man was accused of stealing another's wife and taking her into the Yukon from Alaska. The man was arrested, taken to Fort Constantine -- and promptly escaped downriver out of the fort's rickety stockade.
Before 1896, justice was largely in the hands of miners' committees, theoretically democratic groups of prospectors who in reality operated little above mob rule. They adjudicated disputed mining claims and judged crimes, often under the influence of alcohol. Trials were seen as a form of entertainment in an otherwise bleak life, and while they did successfully administer justice, that justice was uneven at best.
In spring 1896, the NWMP began to firmly imprint itself. On Glacier Creek in the Fortymile country, a miner found he couldn't pay the employees working his claim. He defaulted, and the employees mustered a miners' committee. The committee promptly ruled against the miner and the owners of the claim. They seized the claim and sold it at auction to raise funds to pay the employees. The owners protested to the NWMP, which mustered a large force on July 4 and ejected the unlawful claimant. After a standoff of two days, the miners' committee folded.
This was typical of the time. The NWMP was out collecting mining fees (something that hadn't been done until 1896), seizing illegal liquor and taking over from the committees. Those who didn't care for the NWMP simply decamped over the border to Alaska.
That changed at the end of the year, when word spread about George Carmacks' find.
When news arrived in Circle in January 1897, almost the entire town folded up and left for the Klondike within weeks. The price of cabins dropped to almost nothing, and you couldn't find a dog team for love or money. Prospectors were used to heading out at a moment's notice to the next big thing, and they did so again. By the time the first gold-carrying steamers docked in Seattle, most of the good claims were already taken -- but of course, that didn't stop the rush.
There were scammers aplenty, selling improper gear or equipment that would never be needed. There were fraudsters selling rights to claims that would never pay out. These crimes by and large were never prosecuted (unless the fraudster offended someone with influence ... and was dumb enough to get caught).
Drunks, in town after cashing their pay, would frequently get rolled and robbed blind. There were occasional bank robberies and store robberies and gold robberies, but it was much worse in Alaska than the Yukon, where the NWMP kept things nailed down.
Sam Steele, visiting Skagway in 1898, wrote, "Might was right ; murder, robbery and petty theft were common occurrences."
Now, you have to take Steele's account with a huge grain of salt. He's writing this 20 years after the fact, and using his memoir to puff up himself as if to say, "look how bad things are without me."
Still, you can't argue with the fact that Alaska was a much nastier place to be than Canada.