Motives do relate to preferred sports; if you play sports to test yourself against yourself, chances are this is rooted in Achievement motive. Likewise, if you play sports to beat the other guy, we're talking Influence/Power motive. (Easy test if this is true: You aren't worried if you don't play your best, because at least you beat the other team.) Even Affiliation relates to sports: do you play to socialize with friends?
So why would someone risk their career and reputation?
Some studies have shown that Achievement motivated people take "short cuts," to put it politely--if their goal focus is strongly enough aroused, they will do whatever it takes to get to the goal, including cheat. Remember, we are talking about powerful emotional drives here--they're hard to resist! Similarly, David Winter identified Achievement-motivated American presidents (like Nixon) as more likely to get caught on campaign violations--they try to raise money efficiently rather than legally!
Affiliative presidents, to continue this thread, appoint friends and defend them regardless of their actual capabilities or honesty. In the Olympics, Affiliation could translate into "not letting the team down" or "supporting my country." The Greek reaction to their doping athletes has apparently been a mix of incredulity, anger, betrayal, and disbelief. It's an affront to Greek identity.
And Influence-motivated people are thinking about how they will look in front of all those people, of course.
Sadly, there are many around the athlete who will encourage this. Small countries, particularly, who simply don't have as many powerhouse athletes, are strongly motivated to make sure the few they have can stand up against the hundreds of athletes from, say, the US, China, or Russia. Some writers have suggested that ultimately all athletes will have to take drugs, because otherwise ordinary humans simply cannot compete. Given that the differences between runners can be in the hundreths of seconds, even a tiny improvement could be the difference between a gold medal...and obscurity. Anyone remember the fourth place winner in anything? Didn't think so.
Fortunately, there are a few people who seem to be into it for the fun, like the Jamaican bobsled team. Yeah, it's fun watching amazing athletes, but it is also fun seeing people who are just plain gutsy--at least for the Power motivated, or those with a sense of humor!