What does this joke between Dean Martin and Frank Sinatra mean?

I have been listening to Dean Martin Pandora radio lately and there is a song medley between Sinatra and Martin. During each song they have little quips back and fourth, and there is one that I don't seem to understand

Frank: "Do you have a fairy godmother?"
Dean: "No, but I got an uncle we keep a close eye on."

What is the meaning behind this joke?

In this quip, the term fairy is being used to imply homosexuality. "Fairy" is a somewhat archaic term for homosexuality that today would be considered offensive.

Nels Anderson in The Hobo (1923) said that “Fairies or Fags are men or boys who exploit sex for profit.” The word fairy appeared in the 1870s, and was universally understood by the 1890s.

(From Rictor Norton's "A History of the Word 'Gay' and Other Queerwords")

The joke comes from Frank's use of fairy godmother to mean the fairy tale character, a woman with magical powers who brings good fortune to someone (as in Cinderella), and Dean using the term fairy to connote homosexuality. Dean is saying "I have an uncle that we suspect might be gay, so we watch him closely." The "keep a close eye on" also suggests that homosexuality is something to worry about or police, so the joke reinforces homophobic norms (not at all surprising given the context).

A common meaning for fairy is homosexual (most dictionaries warn that this use is offensive), and that is one of its uses here. The usual meaning of fairy godmother is a magical woman who watches over the indicated person (often seen in children's stories). See http://dictionary.cambridge.org/dictionary/english/fairy for both meanings of fairy.

So Frank is asking if there is someone who (magically) watches over Dean, and Dean is saying that he has an uncle whom the family suspects is gay. When explained this way, it doesn't sound funny at all, but the original is (somewhat) humourous.

It's a simple gay joke. Others have claimed it involves paedophelia and that it is perhaps offensive. No. It's just lowbrow humor from a few decades ago.