The best place to live really depends on what you’re looking for, as well as how much you’re willing to spend.
You can find the best selection of high-end, Western-style apartments in District 2, particularly Thao Dien and An Phu (as some of the other people mentioned). This area is quiet, well developed and offers a huge amount of choice, with lots of apartment buildings for you to select from. Expect to pay around $500 to $1,000+ for a one or two bedroom in Thao Dien, although you may be able to find something for less than this.
There are also a few other good areas for apartments. Binh Thanh District is a heavily built up part of town that has huge apartment developments, such as Vinhomes Central Park and Saigon Pearl. There are also lots of apartments in District 1 and 4, although these tend to be on the smaller side and the areas are quiet busy and noisy.
Finally, if you’re looking for somewhere peaceful and highly developed, it’s worth considering District 7, particularly Phu My Hung. The apartments here are large, high quality and usually have great facilities, with pricing a little lower than in District 2.