They're not actually whiskers. They're whisker marks, just marks on his face that resemble whiskers.
Yes, this has to do with the kyuubi. When Naruto was influenced by Kurama prior to birth, he gained the whisker marks:
Naruto's most prominent physical characteristics, however, are the whisker marks on his face that he gained from Kurama's influence on him while he was in Kushina's womb.
The reason Naruto's mother did not have them is because these are as a result of prolonged exposure (being in the womb) rather than the beast simply being sealed within the host.
Both of Naruto's kids, Bolt and Himawari, have two whisker marks on each cheek, neither of them is a Jinchuuriki but both are the children of Kurama's Jinchuuriki, which implies that this is an inherited trait.
