Sharing the same last name, are Danny Kaye and Mel Brooks related?

Apparently not, at least not on the paternal side.
http://www.brooklyneagle.com/categories/category.php?category_id=12&id=26476

Brooks' paternal family was from Danzig.
https://en.wikipedia.org/wiki/Mel_Brooks#Early_life
Kaye's from Ukraine.
https://en.wikipedia.org/wiki/Danny_Kaye#Early_years

It's interesting that they both were born in Brooklyn and attended (different) high schools there.

Kaminsky, from kamien which means "stone" in Polish (Ukrainian êàìåíü) is a fairly common surname.
http://genealogy.about.com/library/surnames/k/bl_name-KAMINSKI.htm

Hebrew culture uses a patronymic naming system, so many surnames of European origin were adopted fairly recently in order to comply with contemporary custom, often without regard to actual bloodlines.
https://en.wikipedia.org/wiki/Jewish_surnames