Blache et. al, On Approximation Intractability of the Bandwidth Problem, 1997 confirms there is no PTAS for the problem unless $\text{P} = \text{NP}$, even for (binary) trees. Unger W, The Complexity of the Approximation of the Bandwidth Problem, 1998 show that for any constant $k \in \mathbb{N}$ there is no polynomial time approximation algorithm with an approximation factor of $k$. So, unfortunately there's no PTAS nor APX for the problem.
However, for some types of graphs, the problem can be solved or approximated in polynomial time. For a recent survey, see Petit J., Addenda to the Survey of Layout Problems, 2011. In the survey, see Tables 3, 4 and 8. The survey also gives a nice list of references if you want to dig in deeper to some direction. This is an more updated version of the older survey by Diaz et al., A survey of Graph Layout Problems, 2002.
In case you are interested in the exact algorithm as well, I think currently the fastest one is given by Cygan M. and Pilipczuk M., Even Faster Exact Bandwidth, 2012. The algorithm runs in $O(4.83^n)$ time.
