Is the HNF basis the worst basis for a lattice? I am researching lattice problems and some methods for solving them. I read some books that mentioned Babai's algorithm for finding the Closest Vector Problem (CVP) cannot be successful with a "bad" basis for a lattice. 

Which begs a question: what is the worst basis for a lattice?

Is the HNF basis the worst basis for finding CVP in a lattice?
Whether an algorithm succeeds or not for solving e.g. CVP depends on the quality of the basis, which for instance can be measured in terms of the sum of the norms of the basis vectors. Your question could therefore be interpreted as: does the HNF give you the basis of the lattice with the worst quality?

The answer has to be no to this. Suppose w.l.o.g. that the lattice lives in $\mathbb{Z}^n$, and the sum of the norms of the basis vectors of the HNF basis is $M$. Now clearly there can only be finitely many bases of the lattice for which this quality measure is less than $M$, as we can just enumerate all possible such bases. On the other hand, there are infinitely many bases for each lattice, and so there must be a basis with even worse quality. It all depends on how you define "quality", but for most sensible definitions there is no "worst quality" - you can always get a worse basis.

More concretely, given a basis matrix $B$, you can obtain another basis $B' = U \cdot B$ by multiplying $B$ with any unimodular matrix $U$. By taking a $U$ with huge entries, $B'$ will likely be a much worse basis than $B$.