Not very according to this piece in Slate:
Unfortunately, more fresh food closer to home likely does nothing for folks at the bottom of the socioeconomic ladder. Obesity levels don’t drop when low-income city neighborhoods have or get grocery stores. A 2011 study published in the Archives of Internal Medicine showed no connection between access to grocery stores and more healthful diets using 15 years’ worth of data from more than 5,000 people in five cities. One 2012 study showed that the local food environment did not influence the diet of middle-school children in California. Another 2012 study, published inSocial Science and Medicine, used national data on store availability and a multiyear study of grade-schoolers to show no connection between food environment and diet. And this month, a study in Health Affairs examined one of the Philadelphia grocery stores that opened with help from the Fresh Food Financing Initiative. The authors found that the store had no significant impact on reducing obesity or increasing daily fruit and vegetable consumption in the four years since it opened.
and
Earlier research suggesting that better fresh-food access improves diet and would therefore improve the health of people living in poverty was drawn from small samples or looked at store availability in narrow geographical slices—often without information about how or where the people who lived there shopped. “They never link the neighborhood characteristics to actual individuals,” explains Helen Lee, author of the Social Science and Medicine study. “Without that, all you have is speculation.”
Lee also notes in her study that, on closer inspection, food deserts don’t actually exist in the U.S., at least not as a national problem—on average, poor neighborhoods have more grocery stores than wealthier neighborhoods.
The writer concludes on a note which which I agree. The real issue here probably isn't access to fruits and veggies. These are simply symptoms of a larger problem. Poverty. Alas, solving the problem of poverty is no easier or no less complex than solving the problem of obesity. But, if you could made headway on poverty, I'd argue people would have a greater incentive to consider their own future health outcomes.