They definitely won't laugh at you, my bread and butter work is rebuilding taps!
All plumbers charge differently and some taps will need a new insert (ceramic quarter turn type) rather than a washer but if it is simply a washer then it will be their minimum charge as it should take much less than 30 mins.
Plumbers charges vary between £40-£100/hr with some of them charging an hour as a call out charge.
If there were any other 'small' jobs they could do while they were there then you would get much better value for money!
I know how ridiculous this is, but our kitchen tap is very drippy and getting worse. I don't have a clue how to fix it, and am not remotely handy, and DH has been saying he'll try to replace the washer for about six months now, and shows no signs of doing so.
How much would a plumber or handyman charge to come and do it please? and would they laugh at me