I believe red ink is generally frowned upon as hard on the eye. For informal letters, any colour is OK. For formal letters, if written by hand, I would play safe and stick to black or blue.
I would probably avoid using the bright coloursthat are available for a social letter, like a thank-you letter; I'd only use them for friends. Mind you, I haven't written a letter by hand in years. In Britain, some people feel that email is not suitable for thank-you letters, however, and that a letter thanking a host, say, should be written in ink and not word processed.