Republicans have had a significant advantage in doing so, and this probably dates to Nixon's 1968 election. Reagan represents the apex of this effort, and W. Bush was an echo, but recent failures seem to have thrown this effort off course. This is an amazing development considering the fact that Republicans were talking about having created a permanent majority after the 2004 election. Karl Rove's goal was to further peal off key constituents of the Democratic Party and build a communications network that would maintain their commitment to the Republicans for at least a generation. Things do not seems to have worked out as hoped.
The major stories of the Clinton-Obama struggle, to me anyway, has been the way that they have developed organizations in the states and localities that they have competed in, and the opportunities that gives the Democratic Party to redevelop the organizational strength it enjoyed back in the New Deal era. The very fact that the Democratic primary was fought to the end, and forced both candidates to connect with the state party organizations in each of the 50 states gives them an organizational advantage. There is no guarantee of course that this will work to their advantage, but it seems that they have identified where there problems have lied in the past few electoral cycles.
In Texas, organizational redevelopment seems to be the hallmark of their recently reelected party chair Boyd Richie. In Burnt Orange Report's support of his candidacy, Richie's ability to drag the party into the 21st Century was touted as his key contribution to the party's resurgence:
Boyd Richie has accomplished a great deal during his time as TDP Chair, and his leadership has changed the landscape of Texas politics in favor of Democrats in remarkable, tangible ways. Chief among those positive changes are developments in the areas of finance, field organization, technology, and staff.
I have often argued that money in politics is neither good nor evil but absolutely necessary, and the Texas Democratic Party's fundraising has surged over the last two years. The party has increased the number of donors a great deal, taken in significant money on the internet, and initiated a monthly Democracy-Bond-esque small donor program called Majority Builders. The Majority Builder program is precisely the kind of revolving continual revenue system a political party requires to not only keep the lights on but reach for new highs in communications and organizing capabilities. For the TDP this has already proven to be a great success.
Field organizing and the TDP's connection to and engagement of grassroots Democrats all across Texas has also been an arena of extraordinary advance in the last two years, and much of this is directly attributable to Boyd Richie. His marathon tour across Texas last summer, during which he held 18 Town Hall meetings with grassroots progressive activists and party organizations at the county level and below, allowed the party to bridge the somewhat overhyped ideological gap between grassroots activists and party officials. Through that process the party's leadership learned (in a sometimes visceral way) what Texas Democrats wanted out of their state party organization. Many of those ideas translated into real action and outreach on behalf of the party, and a more engaged and strengthened Democratic base augmented an already robust and growing infrastructure.