Patrick Egan expects Democrats will temporarily become more trusted on terrorism:

This is the kind of event that happens only every once in awhile: elites from both parties praise action taken on an issue by the party that does not “own” the issue.  What we’ve seen taking place after similar events in the past is a significant but only temporary dip in the reputation advantage held on the issue by the owning party.

As an illustration, consider the issue of education, where the parties’ issue reputations are in reverse.  Compiling poll results from the past two decades, I estimate the long-run advantage held by the Democrats on education (controlling for any concurrent general sentiment toward the two parties) to be about six percentage points on average.  As shown in the [above figure], the passage of Bush’s No Child Left Behind in 2001 (with the imprimatur of liberals like Ted Kennedy) caused a significant dip in the extent to which the public trusted the Democrats more than the Republicans to handle the education issue.   But this change was temporary: by the end of Bush’s first term, the Democrats’ advantage was back to its equilibrium level.  That’s my best guess about what will happen with regard to the Republicans’ reputation advantage on terrorism in the wake of the killing of bin Laden.