By Danielle Maurer, Anthropology Major & SciFi Fan
Firstly, "Winter Is Coming" are the Stark words. Every great House in Westeros has their words, which usually say something about their character and family values. For instance, the Greyjoy words, "We Do Not Sow," indicate that the Iron Islanders are not farmers but instead do a lot of reaving and sailing and fishing. So in regards to the question details, I think the words do say something about the Stark characters - perhaps not that they are natural pessimists, but certainly that they are mindful of the future and that worse things may lie down the road. Living in the North, the Starks have the most experience of the great houses with harsh winters and they know the value of being prepared.* All of the great houses have a tendency to say their words, though the Starks easily do it the most.
In many cases, the words are simply literal. At the start of Game of Thrones, an extremely long summer is coming to a close - which means that it will be followed by an equally long (if not longer) winter. A winter of several years is going to naturally be a difficult time. There is a sense in which Ned saying this to Cat and Benjen refers to that. The maester at Castle Black acknowledges this later, saying that "The Starks are always right in the end."
Specifically, at the beginning of episode one Ned tells Cat that "winter is coming" after she objects to Bran seeing the execution of the deserter from the Night's Watch. I think he says this to Cat here for multiple reasons.
1.) Bran is a Stark and must learn how to act like one. Taking Bran with him contributes to Bran's education on this point. The Stark words reinforce the idea that Bran must learn how to be a proper Stark. This gets further reinforced later when Ned explains why he personally swung the sword and reminds Bran of their words.
2.) Bran needs to be prepared for what's coming. Life is full of dark times, and the sooner Bran learns about this, the sooner he can learn how to prepare himself against what's coming . It's removing the veil from his eyes. And the fact of the matter is, as I've already said, a long winter is coming - and Bran has never known anything but summer. It's kind of a metaphor for coming into adulthood and losing childhood naïveté.
In the second instance, Benjen and Ned are talking about how the Night's Watch deserter reported seeing a White Walker. Beyond just saying this because Benjen too is a Stark, Ned likely responded "winter is coming" because:
1.) The White Walkers were last seen during a particularly bad winter a long time ago. I think they make reference to this in the show. While Ned and Benjen both seem to initially discount the idea that the White Walkers are back, both of them have enough experience in the North and, in Benjen's case, beyond the Wall, to know that they could be wrong. Ned here could be referring to the possibility that the White Walkers might come back soon with the next winter - especially if it's as bad as it's expected to be after nine years of summer. You can almost read this "winter is coming" as "the White Walkers might be coming."
2.) Ned acknowledges that if the White Walkers truly are coming back, then the Wall (and the North) needs to be prepared. It's significant that he says this to Benjen, because Benjen is a respected ranger of the Night's Watch, charged with the protection of the realm from things beyond the Wall - and obviously, the White Walkers fall into that category. It's a subtle reminder of the duty of both Starks to protect their people from danger. It's a reminder to be vigilant, just in case.
Ultimately, there are plenty of reasons for Ned to say his House's words before anything "bad" happens - especially when I get the impression that the point of saying those words is to be prepared .
And of course, it's also very obvious foreshadowing for the rest of the series.
*Martin himself talks about this briefly in the House Stark Feature:
 Again, I don't think this is necessarily pessimism. I think it's more pragmatic than that. The Starks clearly love each other and have a loving home life (minus the tension between Jon and Cat). If they seem dark and grim, it's because that's how you survive to get the happy moments.
 And to clarify - plenty of "bad" things have happened to Ned that occur before the main story takes place. Ned's sister was kidnapped and possibly raped before dying; the Mad King set his father and elder brother on fire. Ned carries the weight of that knowledge with him; he knows how easily and quickly "bad" things can happen, and often with no advance warning.
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