One of the ropes I used to fix tomatoes loosened and the stem with lots of tomatoes partially broke, but didn't break completely. Should I cut the broken part? Is there some other way to fix it? It has many green tomatoes on it, that need at least 2 weeks to ripen.
That injury looks more like a "sprain" than a break. In other words, the stem is injured, but the tissue isn't completely severed. If you can immobilize it and provide additional support to keep the weight of the ripening fruit from damaging it further, I bet it will recover. (Patient needs to avoid strenuous activity though.)
This happens all the time with tomatoes. Get a stick with a Y formation, stick it in the ground with the Y holding up the tomatoes to a point that that bend is straight again. Most certainly will do the trick. I do this often and the tomatoes ripen just fine. If the Y stick doesn't work you can also use duct tape and a stick like a wilderness survival trick for a broken arm. Won't cause any harm during the short time needed for ripening your tomatoes.
Last winter I wanted to get tomatoes off the vine before the freeze. Take off green tomatoes too soon and they just don't ripen. I've found through experience that if a green tomato has started to change color the tomato will continue to ripen after being removed from the vine.
This change in color might be as subtle as being a little more yellower than the other green tomatoes. There might be a spot starting to color. Then it is fine to pick and put on a shelf not in direct sunlight. Putting them in a bag is well, 'out of sight, out of mind'. I am very bad. On a shelf, they ripen within days. Right in line of sight, I'll grab it and use it or eat it.