"On the reduction of a random basis" by Ali Akhavi,Jean-Francois Marckert, and Alain Rouault
purports to answer your question.
Added later: There is a real question about what is the right meaning of a "random lattice". You should see the paper "On the equidistribution of Hecke points" by Daniel Goldstein and Andrew Mayer Forum Mathematicorum Volume 15, Issue 2, Pages 165–189. In that paper they look at the natural Haar measure on the space of $n$-dimensional lattices.