Star Trek has quite a few of these. Probably the most notable of these is Eddie Murphy, who was offered and almost accepted a role as the whale scientist in Star Trek IV: The Voyage Home. Rather than being a romantic interest he would have been a comedic character, playing a crazed scientist who bumbles his way into being part of the crew's plan to rescue the Humpbacks. Murphy was hot off the heels of Beverly Hills Cop, and while fans seemed opposed to the idea, it almost certainly would have made the movie an even bigger success. With a box office of 316 Million BHC was the highest grossing movie of 1984 and had catapulted Murphy to superstardom. While Star Trek IV was the 7th highest grossing film of 1986 and made a very respectable 109 million, that's still only 1/3 of what BHC did. Assuming Murphy pulls even a fraction of his previous fans to see the film it could very easily fight Top Gun, Crocodile Dundee, and Karate Kid II for a spot in the top three highest grossing films of the year.
This would cause some pretty big butterflies for the franchise going forwards. For one, the lighter comedic tone of the movie is vindicated, and the franchise has re-justified getting blockbuster budgets. Star Trek 5 and 6 are probably very different to the ones we got OTL. TNG will also be impacted. One of the main reasons it was greenlit was do to TVH's success. With the movie being an even greater success, it's probably given a bigger budget and maybe even a proper network release.
A decade later a very similar situation occurred when Tom Hanks was almost Zefram Cochrane in Star Trek: First Contact. Hanks had launched himself to superstardom with the success of Forrest Gump (behind only The Lion King as the highest grossing film of 1994). Ultimately chose the pass on the film to direct That Thing You Do! First Contact grossed a decent 92 million, while Forrest Gump made over 683 million. Assuming Hanks puts some extra butts in seats the movie could do a lot better. Probably not top 5 of the year, but top 10 would be pretty realistic.
This probably doesn't have too much of an impact of the series as a whole. It might temporarily raise the TNG movies from mid budget affairs to high budget ones, but I doubt they could keep that up for long. The Berman Era had already started it's death spiral by this point. If Ronald D. Moore and Brannon Braga get to write the sequels then they'll be at least somewhat better, and Jonathan Frakes probably directs the rest of the series. Assuming ALT!Insurrection and ATL!Nemesis are even marginally more successful there will probably be 1-2 more TNG movies. I'd guess a release in 2004 and one in 2006, giving them six movies like the TOS crew and ending on the franchise's 40th anniversary.
I don't imagine TV viewership numbers would be effected much. Enterprise might be able to squeeze an extra season or two out of itself, but you can only beat a dead horse for so long. The biggest difference here is that 2009 Trek isn't made, as it was only produced because of the failures of the TNG movies. This likely means JJ Abrams never gets his hands on the Star Wars Sequel Trilogy.