Cycling is not seen as being worse than other professional sports, at least not in the US.
NFL Football players have been found guitly of such crimes as carrying a concealed weapon to a nightclub, running a dogfighting ring, manslaughter, spouse abuse and cocaine posession. Others are alleged to have commited rape and murder.
NBA Basketball players are linked to just about every crime you can imagine that involves recreational drug use.
An NHL Hockey player was found guilty of aggravated assault during a game as he hit another player in the neck with his stick. In that sport, players are allowed to fight without penalty (as long as the fight is one-on-one).
MLB Baseball has an even larger steroid scandal on its hands that does cycling. The United States Congress felt compelled to hold a special hearing on the matter.
Professional boxing is thought by many to be completely fixed by organized crime.
College athletic programs are routinely punished for recruiting violations or forgery of documentation to establish eligibility of its players.
In other sports, players routinely break the rules. They are given a small penalty during the game (free kick, penalty yards, player fouls, time in the penalty box, etc.) and then the game resumes and little is ever heard of the fouls again.
So, a few cyclists hung onto a car to make it over a mountain. They weren't in contention for victory. They used bad judgement in order to try to make it to the finish. They were punished for their indescression. I doubt anyone outside of the sport knows of what they did or would even care.
Cycling seems to have a bad image only amongst its fans. When I speak of doping to the casual observer they act as if it were the first time they ever heard of it.
Outside of the sport, cyclists are generally regarded as men with funny hats, strange team names, shaved legs and tremendous endurance. The popularity of cycling has grown an incredible amount and the industries that build and sell bikes are doing well.
As fans of the sport, let's be happy that only one rider has been linked to doping during the sport's marquis event. Many publications have heralded the fact that the TDF this year was the first since 2000 not to be rocked by a doping scandal. The image of the sport is not perfect, none are, but it is improving dramatically.