Central Accommodation in Copenhagen

Copenhagen is quite small and really anything in Indre By will put you within walking distance of everything. I haven't found a neighborhood in that section of the city that makes me uncomfortable or feels sketchy. It's one of the cleanest, safest places I've been in Europe. Last time I was there I was by myself and no matter how late or early I was out and about I felt very secure even though I was staying in an apartment with no hotel personnel around to rely on!
I stayed just a block from the Stroget shopping street and around the corner from Kongens Nytrov square near to the beautiful Nyhavn canal. There is a metro station right on the square so you can get anywhere you want if you don't wish to walk. Anything in that area is convenient and in a lovely part of the city.
To get back to the cruise facility just call a cab...dial 35353535 and one will show up within moments! Easier than the metro with bags.

As Anita says, anything in or around Indre By (Inner city) is central and safe. Generally Copenhagen is very safe, the only area worth mentioning is the area behind the Central Station around Istedgade, which has a lot of discount hotels and is a bit on the seedy side but not at all unsafe. This is the case almost everywhere in my experience, the areas behind major train stations are always a bit on the dodgy side, but have plenty of cheap hotels. The area around Kongens Nytorv is on more posh side and the Nørreport area also have a number of decent hotels.
Btw you don't necessarily have to have access to the Central Station. Nørreport Station is also a major hub for trains and you can get there from the Airport by taking the Metro. Use www.rejseplanen.dk to plan all public transport.

We stayed at the Kong Arthur Hotel last autumn, and it was great. It is a very short walk to the Norreport Station, which as Morten points out, is as convenient as being near the Central Station. Our room was very nice and the price was reasonable. We did not eat the hotel breakfast. If you book directly with the hotel, I think you have free access to their spa too. They also have a free "cozy hour" between 5 and 6 where they offer a free glass of beer or wine. I would definitely stay there again.

I loved my recent stay at Avenue Hotel in Fredriksberg. It was a quick 5 minute walk from the Forum Metro station (direct line from the airport) and close to buses to take you across the canal to the center. Plus, free wine every day from 5-6 PM!

I found getting to the cruise terminal a pain in the patootie via public transport, taxi wasn't that expensive and a heck of a lot easier. I think 40 usd from central train station. Believe me, I researched the heck out of it. We stayed at http://www.absalon-hotel.dk/en/ close to the train station and walked all over town. Easy to get to from airport via train.

Kathi getting to the cruise port (I assume it's the one on Oceankaj) does not have to be difficult at all. There is regular bus service (Bus nr. 27) there from either Østerport or Nordhavn station, it runs every 15 minutes during peak hours and every 30 min during off peak hours. From the Central station it's 35 minutes with train and bus. Tickets are 24 kr.
Unless you have luggage you can't carry or have to be there at super awkward hours, there is no need for a taxi.
The link I posted earlier let's you plan all public transport in Denmark.

Indre for sure, you really don't even need the U there, you can foot everything if your mobile. I had a great airbnb there, Host was Ma, if your interested, PM me and I can send you the link so you can take a look at it,

We stayed in Copenhagen a year ago, we rented a very decent sized apartment on Vestobrogade. 2 people. The bus stop was outside the door, and it was 15 minutes walk to the Central Train Station, 2 Grocery Stores very nearby. an excellent location.