- Is there an objective definition of attractiveness? I doubt it. Probably you're looking at so many dozens or hundreds of responses using a Likert scale to reach an adequately powered sample for analysis. Actually, you might need more than one - do people need to differentiate "cute" from "handsome" or "sexy"?
- Who are you asking: fans or the general public? I wonder if there's a difference. Would fans find a player more attractive because he played for their team, was a great player or won championships? I think that would have to be evaluated before embarking on the main body of research. And then you'd have to evaluate whether or not that held true for fans from all teams or just fans from the player's home city. If you went with fans, you'd likely need a cross-section of fans from all over the country. If you went with the general public, you'd probably need a set of questions aimed at determining if they are or are not fans, and if so, how much of their lives are dedicated to watching basketball.
- In the context of the time in which they played or the modern era? One could try to evaluate how attractive the long-ago peoples of the 1950s found their sports heroes, but that seems likely to turn into some kind of tedious historical sports writing exegesis. I think you have to go with live people you can survey today, but maybe it would be interesting to see if you have to adjust for age. Probably you'd need a question about how long each respondent has been paying attention to basketball, and what kind of time they've put into it over time. That would be a hard question to ask, I think!
- What is a "basketball star"? You'd either need a list determined by a trustworthy third party (like the NBA itself, or maybe something like Sports Illustrated) or you'd have to craft a definition of "star" from scratch. You could go with a Lexis-Nexis search and pick the players who appear the most frequently, but you'd also get players who had a lot of scandals but weren't necessarily stars. You could go by pay scale, but you'd have to determine whether or not well-paid players are always stars and if they're considered as good at the beginning and end of their careers. You could go by stats, but you'd have to determine which stats, at what tolerances, and if your chosen tolerances are appropriate in every decade. I mean, maybe rebounds and assists are important now and there are star players who only score 4 points a night but kick ass at getting the ball into the hands of their teammates, but was that true in the 60s? in the 70s? And did the ABA and NBA have different methods of tracking stats? There are a lot of questions here that I'm sure a sports statistician would be happy to suss out with you.
Do a pre-survey to get basic demographic information (age, gender, self-identified race and ethnicity, level of education, city/state where they spent the most years between the ages of 1 and 18, current city/state, how long they've lived there, socioeconomic status, marital status).
Then show a slide show and ask them to rate the attractiveness of the picture they're seeing. I'd use two or three pictures for each player in random order. You could use two pictures in uniform - one home, one on the road- and one not in uniform. (But which uniform? Maybe wherever they played longest?) It would also be interesting to try putting names on the photos for some people but not for others. How many respondents would recognize all of them by sight alone?
Do a post-survey with your fandom questions and maybe thoughts on attractiveness in general. I'd be sure to ask which team, if any, they rooted for as a kid, and let them pick as many as they like. I'd also ask if they thought someone had been left out of the line up of stars.
You'd have to account for people who don't use the internet. Maybe you could have a study center and advertise publically for people to come in and take the survey in person. That might hurt your national sample, though - too much input from one region.
Analysis: Aside from proving or disproving the hypothesis, there's a lot of potential for exploring some interesting questions about how fandom influences perception. Do the people who self-identify as rooting for Atlanta differ in their perceptions from people who self-identify as rooting for the Hawks? (Yes, I know that's the same team. That's the point: is one person rooting for the city while the other is rooting for the actual team and the players who comprise it?) Also, does it matter if you know who the pictures are of? How many people recognize the names of sports heros without having any idea what they look like? How many people recongnize pictures of sports heros without having any idea who they are? What is the influence of age on ideas of what makes an attractive basketball player? Is there a different standard of attractiveness for basketball players and non-basketball players? What kind of comments do people make as they're making their evaluations?
Okay, sports social science: get to it!