Those really dark (twiggy)canes look dead to me - off with them. There is a long but slender cane coming from low down on the right - shorten by up to half. Would like to see a pic of the top of the rose because there are 2 older canes growing very closely together and it might be worthwhile losing some or all of one of them....but would like to see the top. Other than that, if you wanted to try to get a bit more basal growth, you could try nicking the trunk, just below where it curves to the right - a shallow incision, cutting through the outer bark, about a third of the way round the circumference of the trunk - this can sometimes stimulate a cane to arise, a bit lower on the cane than the cut.
If you could get a pic of it all, there are some people who can do that little arrow thing on the screen, pointing out where to cut. No need to worry though - it doesn't look like it is absolutely desparate for more than a bit of die-back removal....and not for a while either, so you can look at loads of pics and Paul Zimmerman's videos

Thanks so much for the rapid reply campanula - I am adding the only other photo I have (I'll have to take a few more..didn't take many).. cheers!

Hmm, I wouldn't be going mad with the secateurs on this one. However, take a look at the slenderest canes - these need a chance to thicken up so you can cut them back by half.On the furthest branch to the right...which turns towards the horizontal, there is a pale colourred lateral cane which ends in a fork - cut that back to just after the tiny twiglet pointing towards the top...and snip the bit of die-back off the end. I can't see what is happening to the thin centre canes but summat does not look right. In the 3 greyish canes, the one nearest to the left looks like it could be cut back to the newer green cane going off the pic on the far left. However, don't do this yet - I would be inclined to simply wait till March or thereabouts before doing any pruning at all because the winter is far from over.

I was planning to wait until early spring to do the pruning. I have also taken your advice and watched a few of the Zimmerman videos. Thanks so much - this has been enormously helpful!

I'd strongly suggest leaving it alone for the foreseeable future. In the spring, you can cut off dead wood. When it blooms, post pictures and see if we can ID it. It isn't a hybrid tea/floribunda type, and pruning that is inappropriate for the type can do a fair amount of damage.

I'm not sure what zone NYC is (maybe zone 7?) but I think it is too early to prune yet. A very general basic rule of thumb is to start pruning when the forsythias bloom in your area. Usually that means that temperatures are right for the roses to start growing and it's time to prune.

I tried to do the arrows picture but it was very hard to tell from your photos what was good and what wasn't so here is my Basic Pruning Primer

Take off any dead wood. Any cane that is black or very dark brown is probably dead wood. Canes that are green but have a very black or dark brown streak running down the side are probably going to die soon so take them off to below the streak. However, as some canes age they can get a darker brown barky look to them and still be living wood. You can tell dead wood by looking at the center part of the cane called the pith. Start at the tip of any cane and cut off a piece about 2 inches long. Look at the center. If it is a greenish white or cream the cane is alive. If it is brown or a dark tan it is dead wood. You can easily tell the difference. The whiter center will look moist and the dark center will be dry looking. Keep cutting down the cane in small pieces until you reach cane that has a clear whitish moist center. The cane from that point down should be healthy.

Once you've removed all the dead wood you can prune for shape and size if you wish. It's a good idea to take out one of any canes that cross or rub each other. Rubbing will cause damage and you could lose both canes so take one out.

Take out any spindly canes with a diameter of less than a pencil. You have a lot of that spindly.twiggy side growth on your bush. Also take out any canes that grow directly into the center of the bush. You want to open up the center for good air flow. That allows the center to get more light and to dry quicker to help prevent fungal diseases. Since your bush doesn't appear to be very large you may want to hold off on thinning it too much right away though.

I am not at all familiar with the method that Campanula described for getting it to put out a new cane at the bottom (basal growth) but Camp knows her stuff so I'm sure it's good advise. I was going to suggest that if possible you repot it deeper to encourage more growth. You can also then straighten it up in the pot some so you don't have to stake and tie it like that. My worry is that eventually that string is going to strangle the rose and kill it. It's wrapped very tightly around that one main cane. If it girdles the bark completely everything from there up will die.

Most of all don't be afraid to make mistakes. For the most part you can't really kill a rose by pruning it. Usually it just grows back even better. And for another thing, we've all made mistakes and still do sometimes when pruning a rose. I experience that sick feeling of "OOPS, I shouldn't have cut that" usually at least once every pruning season, lol! Mistakes are how we learn sometimes and, like I said, roses are very forgiving.

I know there are several rose societies in NY and I'm sure if you contact one of them they would be very happy to help you out. Or you can go on the American Rose Society web site and find a consulting rosarian for your area that you can contact. When I did that for my area the guy came right over to my house to help me and we've been good friends ever since! Rose people are really nice and love to share and talk roses with anyone interested, lol!

Here is a link that might be useful: The American Rose Society

bartleby1, I live in Brooklyn which as you know is not far from nyc. The USDA has revised its zone hardiness map so that we are now both in zone 7. It's too soon to prune now, so hold off until end of March/early April. Once you prune the rose will want to start growing so it's best to do that when the outside temps are warm enough.

Are there any rose nurseries near you that you can recommend? I'm looking for a new source as the choices in my neck of the woods are disappointing.

This is from someone who knows zip about roses sooooo, anyhow, may I suggest you take a few post-it notes, number them and stick them on the branches, then retake your pictures. That way Camp and others can direct you to the correct cane to cut. ie. above # 6 about 2 inches cut at a slant so etc etc.....