Unless the Scouts change their organizational view with respect to gay members and atheists, I don't think I'd encourage my sons (should I have them) to do it.
If the Scouts don't want to associate with gay people or those who don't espouse religious faith, it doesn't seem to me that the organization is willing to leave behind a WASPy 1950s worldview. That's not realistic, and not really relevant anymore. Sure, things like being responsible and honest still are, but the BSA doesn't have a monopoly on teaching those values. I do think the reluctance to change is what'll continue to cause a decline in membership, and that's what'll devalue what it means to make Eagle. Worse, it'll be seen as an organization that fits on one side of the culture wars (if it isn't already).
I think the role of the antigay policy in driving away members is overstated. If that was the main reason why the BSA is losing members, then the Girl Scouts, who have allowed lesbians in for the past decade, would be seeing an increase in membership. Instead, the Girl Scouts have lost more than half a million members since 2003, and annual donations are down by almost $45 million from where they were 5 years ago. In both cases, I think the lifestyle change that kids these days are going through (more ECAs, more working parents who don't have time to volunteer, more alternative activities available) is the real culprit, and a controversial BSA policy is just a more convenient scapegoat. Plus, while the decline and fall of a large organization that failed to keep up with the times is an oft-told story, one with an issue as inflammatory as this is not, and that's why it keeps showing up on the evening news, regardless of whether or not said policy is the driving force behind the trend.