Usually when an INFP is down, they are down because of a specific aspect of their life. An INFP isn't going to get down by something that is easily fixed, like there is no milk in the refrigerator. INFPs hide their discouragement (although they may offer hints that they are discouraged in a casual manner) while they try to figure that specific aspect out. By the time everyone else notices that they are down, they have been down for quite some time and have spent a lot of time thinking about and trying to resolve whatever is bothering them.
It's natural for people to want fix whatever is bothering them. This usually just frustrates the INFP because they've spent so much time thinking about all the possibilities, and whatever fix is being suggested has already been considered or tried. Sometimes there is no quick fix for things like getting a job in a bad economy, having health issues, trying to find a significant other, money issues, relationship problems, a death of someone close, etc… Often the person trying to fix the INFPs problem will just get frustrated because "they aren't cooperating" by being fixed, and then the INFP gets frustrated because now they have to make that person feel better (since the natural tendency of an INFP is to be there for others). It causes the INFP to feel even more isolated as they withdraw more, because in addition to the aspect that they are frustrated with, they now feel nobody understands how they feel.
So how to help an INFP feel better.
An INFP strong point is being a good listener. I often joke that I've had bartenders tell me their problems. But sometimes an INFP needs what they give. Since this is often isn't a natural ability for others, the best thing is to just try to mimic the behavior.
First, never underestimate the power of listening. For such a passive thing, it takes an amazing amount of patience. Expect that they won't initially allow themselves to talk about their problems. Don't bully them into talking about it. They probably feel embarrassed or stupid for being down in the first place. Just treat it as an exchange. Ask small questions. Consider less about the response and more about if they are responding. If they aren't yet ready to respond, then just give some more time. Don't show frustration. Don't leave the conversation. Don't start discussing a big topic which will be hard to change. The person probably wants to talk but is unsure at this point. I've had people say "I don't want to talk about this", and I say ok, and then talk about small unrelated things, and soon after I'll get a rush of discussion from them about what they said they didn't want to talk about. It's just a matter of making them feel comfortable.
Now this is the important part. Once they start talking, just listen. Be attentive and supportive. Ask clarifying questions. Don't judge or try to offer quick fixes. Afterwards, you might think you have done nothing, and you certainly haven't fixed the problem, but the person will feel better. If it doesn't seem that way at first, then they are probably still emotional from talking about it and just need some time to relax. But the fact that someone cared enough to acknowledge their pain will mean a lot to them.
If you have done a good job listening, you will probably understand why the quick fixes you initially wanted to offer are probably not very helpful. For example, if they are having health issues, saying "you should go see a doctor" is probably a dumb thing to say and is kind of insulting to them, and if you listen for a while, you probably will find out how extensively they have tried to find a cure. But after listening to them and giving them a chance to calm down, you might know of a specific person that is good with this issue, in which case you can offer things like this as suggestions.