الجواب حامدا و مصلیا
If the barber doesn’t provide unlawful services (like cutting beards more than the measure of a fist or shaving it off totally) then undoubtedly his income is lawful. But generally, a barber’s income is from both lawful and unlawful ways of hairstyling and shaving other’s beard, hence his income is mixed with lawful and unlawful earnings. So if there are more lawful earnings in his income then his income will be lawful, and if there are more unlawful earnings then his income will be unlawful. But it is incorrect to say that the income of a barber is unlawful.
And Allah knows best.