It never really fully stopped, though it was increasingly frowned upon towards the end of the Korean war. Probably due to the rise of feminism globally.

Let us first examine where the tradition comes from. Painting aircraft with distinctive markings started in WWI, probably out of a need to be able to identify the different squadrons, they usually consisted of extravagant squadron images, and usually were not individual in nature.

The practice of painting pin-ups, and other nose art, stems from WWII. An explanation for this can be found in the bond that crews formed with their aircraft. A good dose of superstition came with the high casualty rates amongst aircrew, and led to a worship of sorts of the machines that "kept them safe". Of course these crews were away from home, their girlfriends fiancee's and wives, and comprised mostly of young men, so it is no surprise that a lot of sexual metaphors were used. To quote Bomber captain John Muihead on newly arrived replacements who described them as: "trembling virgins waiting for the first touch on their flanks." These kind of metaphors can be seen in a lot of bomber crew's memoirs. A lot of crews referred to their aircraft as a she, something still prevalent amongst pilots and crews even today. The metaphor bombardier Moritz Thomson used to describe the feeling dropping Bombs gave him is similarly striking, comparing it to:"the moment of ejaculation".

It is furthermore worthwhile noting that the survival rate amongst bomber crews was possibly the lowest in WWII when compared to the wars that followed, due to the largely asymmetric nature of conflicts such as the Korean and Vietnam wars. Survival rate of a WWII bomber crew was around 25%. As a result crews grew close to each other as they depended upon one another for survival. Equally so to their machine. This led to the practice of individualizing their machines by naming them, like you would give nicknames to your comrades, and painting it with cartoons. It was allowed and endorsed as a means to promote crew cohesion and morale in a bid to enhance crew's mental resistance to combat stress.

Ann Elizabeth Pfau describes the relation between crews and their machines as being something akin to a mother complex:

"Heavy bombers were the ultimate phallic mothers. Often decorating planes with women straddling or holding bombs, airmen imagined them to be both powerful and nurturing. Their dependence on the aircraft for heat and oxygen certainly heightened this confusion of machine and mother."

We have now established the origin of the practice, let us know turn towards it's decline.

As I already noted the losses in the conflicts following WWII amongst aircrew were much lower, and with the advent of the Jet age crews grew smaller as well. These trends combined probably led to a lower relative level of fear and combat stress. In addition the attitude to portraying women changed, the nudity that can be found in the pin-ups of WWII warbirds was no longer tolerated. Probably as a result of both reasons the practice became less widespread, though never entirely disappeared.

Today some USAF aircraft still have nose art, and Strategic Air Command actually officially allowed them in the Gulf and Afghanistan wars and after, providing they were non-nude.

http://www.airliners.net/photo/USA---Air/Boeing-KC-135E-Stratotanker/0616770/L

Sources

Pfau, Ann Elizabeth. Miss Yourlovin: GIs, Gender and Domesticity During World War II. New York, N.Y.: Columbia University Press, 2008. http://www.gutenberg-e.org/pfau/acknowledgments.html