when i lived in turku, there was a place called daisyladies. it was for immigrant women, and they offered this service.
although nobody was using it; i was their first and maybe only real customer, but it was dead cheap to get some clothes repaired.
so, maybe in helsinki area you can find similar places through the hyväntekeväisyys point of view, voluntary association for the common good...
martat come to mind... but there's also normal commercial repair tailors, although maybe not as common as elsewhere.