Its commonly reported that only 500 people are required for genetic diversity without needing to track inbreeding. Others say the math isn't right and it's actually 5000, but either way the colony is good.
In the 23rd Century they'd be good in any case. I'm sure the medtech exists to repair bad gene stuff, and for diversity they could always ship in some donated sperm and eggs to add some genetic variety. Assuming more people didn't just move there.