If you're looking for a decent and not-too-expensive hostel, I can recommend the Stadium Hostel, which is located in the Olympic Stadium (across the road where the Opera House is located). It's not too expensive, the rooms are nice and clean and security is good.

Budget accommodation in Porvoo

Gasthaus Werneri
Adlercreutzinkatu 29
Tel. +358-400-494876
Basic guesthouse, 400m from the bus station. No breakfast, but free use of kitchen facilities.
€35/50 single/double.
Porvoo Camping Kokonniemi
Uddaksentie 17 (1.5km from city centre)
Tel. +358-19-581967
Camping and caravan ground plus 10 cabins for up to four people. Showers, sauna, kiosk, BBQ shelter, playground. Open June-August only.
Cabins €58-67, camping €12/space plus €4/person.

Porvoo Hostel
(Porvoon retkeilymaja)
Linnankoskenkatu 1
Tel. +358 19 523 0012
HI-affiliated hostel in a Porvoo-style red wooden building. 33 beds in 10 rooms.
Single/double €29/38, €2.50 discount for HI members.