I believe during the Clinton era, but their support wasn't truly locked in until 2004. Clinton left a positive mark on young voters during his second term, but demographic and other changes really began to tilt that bloc towards Democrats in the mid-2000s. Now the weight of racial minorities and a substantially less Republican-leaning generation of white voters have created an age group largely hostile to Republicans.
At least according to exit polling (which are not gospel on this sort of thing but generally backed up by poll subsamples at the time so reasonably believable), Bush narrowly won 18-25-year-olds in 2000. Thus, they were marginally more Republican than the nation as a whole, as Gore overall won the popular vote very narrowly. There is generally no evidence that younger voters were voting more for the Democrats than older voters in the 90s or earlier as well.
The shift to young voters supporting Democrats much more strongly than the nation as a whole first manifested itself at the presidential level in the 2004 election, when Kerry won younger voters by about ten points while Bush won the popular vote overall for a net deviation from the nation of about D+12, far greater in either direction than had ever been the case before. That margin expanded dramatically again to closer to D+40 in the 2008 election and has not materially changed since then.
The causes are complicated. The stronger socially conservative turn of the Republican Party generally during the Bush administration (including especially the same-sex marriage fights), increasing diversity among younger voters (although even younger whites are more Democratic than whites overall), the Iraq War, 9/11 and its polarization of international affairs, the financial crisis, the general incompetence and, in its later years, deep unpopularity of the Bush administration and strong political messaging by Barack Obama surely all played at least some role.