Starcraft 1 was told in a very different manner than Starcraft 2, not the first of which is that the player character is an actual character unto themselves, actually playing as Artanis in the first Protoss campaign. (Compare this to Starcraft 2, where the player piggybacks with Jim Raynor)
In part because of this, your information on events is fragmented - You can't know if Jim and Sarah are in a relationship, because you are neither of them (and they certainly don't confide in you). But to answer your question, there is no strong evidence availible. In-game dialog hints that, if not romantically involved, they will be soon, but, again, nothing is explicitly known.
As to why Starcraft 2's Jim / Sarah relationship seems so different, there are two main reasons. The first is that we are, for the first time, seeing things from Jim's perspective (which we didn't get in the first game), and second, it's been 10 years since starcraft 1, and Blizz has a new crop of writers handling the story.