If someone is sick or sad, I tend to want to baby them up with food, a little present (not expensive) that is specific to them or to their tastes (usually unexpected) and give them an opportunity to talk. I usually would ask questions just to clarify the situation to give them a chance to talk, but don't give advice unless it is clear that they are seeking that or the initial upset is gone and they are now looking for resources/perspective/ideas. Depending on how close I am to the person and how receptive they are, I probably would show some kind of physical affection towards them.
I usually need a chance to vent off the biggest part of my upset and then talk enough that I have clarified in my own mind what is bothering me most. I use the reactions of those around me to also try to calibrate whether my response is reasonable or whether I just need a bit of time to see things in better perspective. Then I prefer to solve it myself or discuss my possible solutions with someone whose judgement I trust.
I don't like it when Te users either try to tell me why I am overreacting and my problem is not that serious, that things will be fine or when they jump immediately to solving my problems for me without understanding the whole situation. I now do realize that they don't mean it to be offensive and are just doing what they would appreciate themselves.
I think often Te users feel helpless if they can't do something active. They don't realize that simply by listening, they are performing a very valuable service for me that I can't do by myself. It helps me get rid of the excess emotion that is obscuring my perspective, it moves me towards finding a solution more quickly, and the act of talking about it with a real person (even if they are only receptive and not giving feedback right away) helps sort the problem and solution out in my own mind.