We stayed at the First Mayfair as well. One night each before and after the cruise. The club across the street did not bother us at all. However, my brother in law arrived very late on the night before my sister and I got back from our cruise (we met up for a weekend in Copenhagen after the cruise) and he came across some ladies offering their services to him. He is from Berlin, it happens there as well in some areas, and he was not shocked. But I thought I should mention it. The other night when my sister and I walked around that corner, we saw a lady who looked like she might be there to offer such services, but she just ignored us. It was not at all in your face in any way. I am sure that in my younger years I would not have had a clue that these ladies were doing anything else than standig on the street waiting for someone. And it was not as if there were so many of them, only one or two.
Otherwise the area felt very safe to me and there were plenty of people visiting the bars and restaurants in that area as well.
The hotel was good, our room was a bit run down, but very nicely decorated. The service was good and the breakfast was very nice, too. For the price (considering other prices in Copenhagen) it was absolutely worth it.
We took the bus from just up the street to the port (had to make one change) and paid about 4$ per person for the trip to the port. Cheap and convenient.