I believe need to create boolean mask and then append value by DataFrame.loc:
mask = (df['col_A'] == 1) & (df['col_B']!='None')
#if None is not string
#mask = (df['col_A'] == 1) & (df['col_B'].notnull())
df.loc[mask, 'col_new'] = 'col_new'
Sample:
In column are strings Nones:
df = pd.DataFrame({
    'col_A': [1,1,2,1],
    'col_B': ['a','None','None','a']
})
print (df)
   col_A col_B
0      1     a
1      1  None
2      2  None
3      1     a
mask = (df['col_A'] == 1) & (df['col_B']!='None')
df.loc[mask, 'col_new'] = 'val'
print (df)
   col_A col_B col_new
0      1     a     val
1      1  None     NaN
2      2  None     NaN
3      1     a     val
In column are not strings Nones, then use Series.notna:
df = pd.DataFrame({
    'col_A': [1,1,2,1],
    'col_B': ['a',None,None,'a']
})
print (df)
   col_A col_B
0      1     a
1      1  None
2      2  None
3      1     a
mask = (df['col_A'] == 1) & (df['col_B'].notna())
#oldier pandas versions
#mask = (df['col_A'] == 1) & (df['col_B'].notnull())
df.loc[mask, 'col_new'] = 'val'
print (df)
   col_A col_B col_new
0      1     a     val
1      1  None     NaN
2      2  None     NaN
3      1     a     val
Also if want use if-else statement numpy.where is really helpfull:
df['col_new'] = np.where(mask, 'val', 'another_val')
print (df)
   col_A col_B      col_new
0      1     a          val
1      1  None  another_val
2      2  None  another_val
3      1     a          val