Just what makes a minor race a minor race? Either an inability to expand as fast as a major race, or an inability to keep up with military, social, or research production. Or maybe because either they didn't get the blueprints to the hyperdrive. Or even possibly that they're not viewed as a major race by anyone else, so they have no influence, just the ability to resist other's influence. With that, I see several possible ways to handle this:
1) Have a racial limit that for the major races is effectively infinite that states that the government for race X isn't developed enough to manage more than X planets. Maybe have a line of techs that can't be traded that increases X, so that a minor can work on becoming a major. This would hopefully cause minors to be more careful where they place their colonies, since they're limited to how many that they can have.
2) Same idea, but where planets can't be farther than Y away from the home planet. This would mean that a minor could pack all the planets within a 1.0 or 2.0 range from the homeworld at the start, and they could research to expand that from there.
3) Rather than trying to restrict where the colonies can be, or how many, limit the range of ships to distance from homeworld only. Or maybe distance from homeworld or colony with population of at least X. Again, allow minor-only techs to reduce X.
4) Reduce the racial abilities points available to the race. I dislike this idea, I don't want them to be pushovers, just want them to feel minor.
5) Don't grant them influence. Not sure if this would make them feel minor. It sure wouldn't when they're pouring a stream of colony ships through my space.
Of these, i think I like option two the best.
In GC1, originally, minor races were restricted to their starting sector. This was too limiting, so it was lifted. I'm not sure how the balance was maintained there, but while you'd occasionally get a fairly large minor, most of the times, they'd span 2-4 sectors.
I had one game last night where right away, I had 3 colony ships streaking through my territory, and another one every 4 turns or so. I finally explored that direction and jumped on the PQ 18 planet in the minor's system that it had been overlooking, apparently because it was too busy trying to find planets in my space.