Lattice generation from basis? This might be a very short very obvious answer, because I've yet to come across a question similar to mine in my searches.

Given a lattice L, with a good base B1 and a bad base B2, what stops an attacker from simply re-creating the lattice structure from the public base B2?

It might just be a terminology mix-up on my part, but that's what I recall being the purpose of basis's in regular algebra.
I'll turn my comment into an answer:


  Given a lattice L, with a good base B1 and a bad base B2, what stops an attacker from simply re-creating the lattice structure from the public base B2?


There's nothing stopping them, the lattice structure isn't secret. Like you say, examining the lattice structure is trivial. The use of lattices in cryptography derives from the fact that even if you know the lattice's structure (i.e. have some basis B), deriving a "good" basis is hard.


  Still, deriving the lattice structure wouldn't mean that one could, for example in a CVP problem, easily output the closest vector?


No. :) When you visualize this problem in low dimensions, like 2, it seems like it should be that way, no? But in high dimensions, this problem is very difficult when you have a random (i.e. bad) basis.

Try out your technique. Suppose we have $n$ dimensions. Our basis $B$ is a bunch of vectors going in random directions. We want to enumerate the points close to $\mathbf{t}$. How many points could potentially be "close" to the target? How do you generate these points efficiently with your basis $B$?