In Jesse James: Last Rebel of the Civil War, T. J. Stiles provides a vivid description of an 1873 train robbery carried out by the James-Younger gang.

The Gang knew that every Monday the railroad sent the transcontinental express shipment full of money on this line. The men selected an out of the way part of the tracks with a tight turn, as this forced the engineer to lower the speed of the train. Prior to its arrival they removed some of the railroad spikes and rigged up a rope so that upon command they could cause the rail to jump out of alignment right when the conductor would be looking at it, so as to ensure that he would attempt to stop. They did not want everyone on the train to die in a crash. The location was also a cut where the train went into a kind of trench that had been dug out of the ground so that it did not have to go over a hill. This would provide good line of sight and fire into the train.

Around 8:30 that evening the train came, they sprung their derailment trap, and began firing at the train. Though it had already slowed to 20 mph, it still was going too fast and derailed tipping over the engine and coal car. Half of the gang stood guard and fired warning shot at anyone who showed their face, the others , in full KKK garb, went into the car to extract their prize. As they moved through the train one of the gang members reportedly declared "We're robbing the rich for the poor. We are Grangers. We don't want to hurt you; we're going for the express car." The James brothers and Cole Younger threatened the two Expressmen guarding the mail and luggage until they provided them with the key to the express safe. The did, and the gang stole $2,337 - less than they hoped to find. It turns out the large cash shipment was sent the night before. The three and a half tons of gold and silver bouillon also in the car (easily worth hundreds of thousands of dollars) was far too heavy to make a hasty escape with and would be difficult to pass off without attracting attention. Each bar weighed about 30 pounds and selling it would require re-casting the bar to remove the identifying seal. The coins would have attracted similar attention.

Contrary to popular belief it is doubtful robbers ever leaped from running horses onto trains. Trains would generally go as fast as possible and could easily outrun a horse. Further, they risked being fired upon by the expressmen or engineer who were often armed with at least a pistol. It was far safer to stop or derail part of the train to rob it of a few thousand dollars. While the robbers rarely wanted to harm the passengers (though they also showed them little regard), they did not care about the train and a crash might disorient any defenders. Most robbers also relied on a network of willing accomplices to fence goods and help them move around. The James-Younger gang relied upon the same network that supported them when they were Confederate raiders in the area. The James-Younger gang did not have any need or desire to shoot bystanders, and they made off with enough to not feel the need to resort to petty robbery, though it did sometimes happen.

Though I don't have any statistics on how common this kind of robbery was, the section ends with a news clipping that stated this new form of crime was becoming "alarmingly frequent."

"Common" is a relativist term and probably contributes to your lack of responses. I'll give it a stab and a pun and say they were relatively common. There were certainly more train accidents than robberies between the first great Western train robbery in 1870 and the last (in 1933) but they were certainly less prevalent than the glamorized depictions of the pulp novels and (later) movies would have you believe.

Take the place that might as well be the train-robbery capital of the West: Stein's Pass, New Mexico.

This was a tiny town in far southern New Mexico ─ you know, that little bootheel that juts south, preventing it from being a perfect square? In any event, this was a town on the Southern Pacific railway line, and in the late 1880s and through the 1890s, it was the scene of at least four and as many as eight train robberies, according to the records of John Bossenecker, who wrote about the place in the journal Western History.

There's a lovely account of one robbery in True West Magazine, and you can read a contemporary account in the Sacramento Daily Union.

You see, train robberies didn't happen just anywhere. They required a combination of factors to make them likely. I've written before about means, motive, and opportunity, and the same thing applies here. There had to be desperate men (train robbers were usually men), the train had to be worth robbing (carrying refined metals or payroll), and it had to be in terrain that made a robbery (and escape) easy.

Stein's Pass fulfilled all those requirements. In the last decades of the 19th century, it was part of a mining boom that included southern Arizona and portions of New Mexico. Silver, copper and gold mining flourished, and those mines needed railroads to carry refined metals outward and cash payrolls inward. Stein's Pass was near New Mexico's border with Arizona and it wasn't far from Mexico. If a train robber could reach the Mexican border, he would be guaranteed safety from pursuit by American law officials. Stein's Pass was in a mountainous area, which meant evading the law was thought to be easier. As the accounts I've linked indicate, however, that wasn't necessarily the case.

Furthermore, popular culture tends to remember train robberies both as a Western phenomenon and one limited to the 19th century. That isn't the case at all. Over in /r/100yearsago, I just posted an account of the robbery of the express train between Chicago and New York City, which took place 100 years ago this month. It was by no means the only robbery east of the Mississippi, but it comes to mind immediately as an example.

You ask about the robbers' likely goal. It wasn't the passengers, of course. It was what the train carried. Take the first great train robbery of the West, which took place west of Reno, Nevada in 1870. The robbers in that case were after more than $41,000 (contemporary prices) in gold coin kept in express boxes. The idea of hijacking the train for a short period is somewhat silly and likely borrowed in popular culture from the Civil War experience of "The Great Locomotive Chase" rather than actual practice in robberies.

The more common modus operandi was to encounter the train while stopped or slowed, force the engineer to stop entirely, crack a safe or mail cargo through force or coercion, then make your escape on horseback.

I love this question! Additionally, was there a higher probability for "train robberies", as it were, to occur before or after the connection of the east and west in Utah? Was the connection of the coasts a big factor in railway crime?

Since there aren't direct "answers", that raises a different "meta" question for me:

How would a professional/academic historian go about researching and answering this question?