It's not the nicest way to find accommodation, but if you are on a budget Omena Hotels (they have two locations in central Helsinki) might be a good option. It's based on self-service so there is no live reception, you enter the hotel via a pin code given to you.
The cheap prices especially around New Year's time probably will attract a "party crowd", so probably not for those travelling with families, but still, affordable option for those willing to look beyond the possible negatives for sure. It's probably a better alternative to Finns than foreigners as in the case of a problem (door code doesn't work etc) it's easier to communicate with a person on the phone when you share the language.
Two times I've stayed (not in Helsinki though) rooms were clean. I didn't stay the night, though, but I didn't hear any later complaints of noise from those who stayed the night.
http://www.omenahotels.com/