Which is right?
There is a genetic component to anxiety; some people seem to be born worriers. There is a genetic component to anxiety: some people seem to be born worriers,
This is all fairly pedantic, but the second clause appears to follow directly from the first, in which case a colon is more appropriate than a semicolon. However, the word "seem" confuses the issue, and "some people are born worriers" might be better.
My personal preference would be to make them two separate sentences. I think it's one of those cases where there's no right or wrong. It's how it reads and all work fine.
The two parts could both be sentences, so I would use a semi-colon. If the second part had been a list, I would have used a colon. E.g. "Thare are 4 countries in the UK: England, Scotalnd, Wales, Northern Ireland".