There are some options for low budget accomodation in Helsinki, and they are mostly 1-star hotels and hostels. Also note that you are usually not required to compromise on comfort, and the low budget mostly is just reflected in the lack of services.
Academica Hostel (Hietaniemenkatu 14, +358 (9) 1311 4334, www.hostelacademica.fi) offers private rooms as well as dormitories. Dorms from 25 EUR/person, private rooms from 50 EUR/night.
More information on Academica Hostel at Booking.com
Eurohostel (Linnankatu 9, 09 6220 470, http://www.eurohostel.fi/) is one of the biggest hostels in Scandinavia, situated in Katajanokka, slightly east of the city centre. Also has an on-site restaurant. 135 rooms in total. Single rooms from 43 EUR/night, triple rooms from 24,30 EUR/person/night.
More information on Eurohostel at Booking.com
Hostel Suomenlinna (Suomenlinna C 9, 09 684 7471, http://www.snk.fi/suomenlinna) is located on the island of Suomenlinna, which means limited means of transportation. Dorms from 22 EUR/person/night, double rooms from 55 EUR/room/night.
More information on Hostel Suomenlinna at Booking.com