I think you need assign boolean mask, instead == is possible use eq or use assign:
#if need True False values by condition
stringX = "string1"
df['A'] = df['B'] == stringX
print (df)
A B
0 True string1
1 False string2
2 True string1
3 True string1
df['A'] = df['B'].eq(stringX)
print (df)
A B
0 True string1
1 False string2
2 True string1
3 True string1
df = df.assign(A=df['B'].eq(stringX))
print (df)
A B
0 True string1
1 False string2
2 True string1
3 True string1
#if need values of column by condition
df.loc[df['B'] == 'string1', 'A'] = df['B']
print (df)
A B
0 string1 string1
1 NaN string2
2 string1 string1
3 string1 string1
#if need scalar by condition
df.loc[df['B'] == 'string1', 'A'] = 1
print (df)
A B
0 1 string1
1 NaN string2
2 1 string1
3 1 string1
#if need if else condition with 2 scalars
df['A'] = np.where(df['B'] == 'string1', 1, 2)
print (df)
A B
0 1 string1
1 2 string2
2 1 string1
3 1 string1