My first time playing a new generation of pokemon games I always use my starter, but once I've familiarized myself with the new pokemon available in that generation I tend to box my starter as soon as I catch the first permanent member of my team and never touch them again. Sometimes they manage to stay on as an HM slave though. For the remakes (fire red, leaf green, heart gold, soul silver) I don't really use the starters because I know the gen 1 and 2 pokemon very well from the originals anyway, and have already used al of the gen 1 and 2 starters in their debut games.
Occasionally I will use a starter when I feel they will well work on my team though (for example my sapphire team still uses my starter). But those cases are rare.