Katajanokka is technically an island, although only a narrow canal separates it from the rest of the city centre. The quays here are the first berths in Helsinki after the island fortress of Suomenlinna.
Nearby is Katajanokka Terminal. This actually serves scheduled ferries sailing to Stockholm and Tallinn, not cruise passengers, but it is open daily from 8 in the morning to 9 in the evening.
Tram number 4T starts from Katajanokka Terminal. Senate Square and the Lutheran Cathedral are six stops away.
Trams into the town centre are fast although, if your cruise ship arrives soon after a ferry from Stockholm or Tallinn, they may be crowded.
The obvious alternative is to walk. You can easily see where the city centre is, marked by the green copper domes of the Lutheran Cathedral. Several parallel streets leading towards it.
On the way you'll pass the red brick Orthodox Cathedral with its gilded cupolas.