df.dropna(subset=[1, 2], how='all')
Or
df.dropna(subset=[1, 2], thresh=1)
In [30]: df.dropna(subset=[1]) #Drop only if NaN in specific column (as asked in the question)
Out[30]:
0 1 2
1 2.677677 -1.466923 -0.750366
2 NaN 0.798002 -0.906038
3 0.672201 0.964789 NaN
5 -1.250970 0.030561 -2.678622
6 NaN 1.036043 NaN
7 0.049896 -0.308003 0.823295
9 -0.310130 0.078891 NaN