There aren't really any areas of Johnson City that would be so undesirable; especially when compared to "undesirable" areas in larger cities.
That having been said, you would probably want to steer clear of some areas near downtown (Carver, Carnegie - these are the areas around Watauga and Unaka Avenues) and Keystone. Just below all that, Northside is pretty rough too. Basically anything east of Tree Streets for a few miles until the city gives way to suburbia/countryside.
Just my opinion, of course.

A lot of people don't like living around Antioch because of the large number of ETSU students in apartments, but parts (closer to Antioch Baptist) are very nice.