On countless occasions I have heard "I will be there for you no matter what" and had those words turn to dust as soon as they were spoken, so it's understandable why I tend to brush them off. I don't want to hear those words. I want you to show me that you will stick by me in the tougher times, and you will continue to be there even if it becomes more difficult. The best parts of life are the hardest won, whether that means great friends, great lovers, great passions, etc., and boy, they've earned a permanent place in my heart for all they've done.
It's easy to push people away when the going gets tough. In fact, once a boyfriend has flat-out refused to help me out in an extreme moment of pain--and I couldn't believe it, but it was then when I really knew that nothing would work out between us. And later, he told me, "I wish I had been there." Well, sorry, you weren't. No one can take that back. Those who were always there for me, however, are of so much more value. Even if what they've done for me sounds small, it means the world to me.
One time that stands out to me is when I told my friend I didn't want to eat or drink--or even have the energy to get out of bed to do so, and she came over after a long day with some of my favorite foods. I didn't have to ask her; in fact, I didn't expect her to do a thing at all then. At that moment I felt so cared for and loved, and I really needed to feel that. Or the countless number of hours we've talked on the phone even when we were worn out and beat.
It was never any singular act that told me "I care about you" or "I love you" or "I want you to get better," but that all of them were present in small ways on a consistent basis. I could rely on them to be there even when so many people hadn't, and I could trust that they would stick around even when so many people had disappeared on me. That is huge in and of itself. You can be that person in someone's life, you can be present and stick around--that means so much more than you could ever know to those who are used to otherwise.
And that can be enough. There's no need to push your own opinions as to how depression/anxiety should be dealt with--there's absolutely nothing they haven't heard and/or tried before. Just listen and ask: how can I support you? However, you are allowed to disagree with their thoughts born of depression if you do have a valid counterargument. For instance, I could say: "everyone leaves me," and you could reply, "the people who matter don't leave" or "you're not getting rid of me anytime soon" (both of which have been said to me).
Everything is more complicated than it seems, however. Even if you're their closest confidant, there are words left unspoken and feelings kept hidden, so never assume that you really understand what's going on or even who they are. The answer to how to support people you care about is a simple one, but oftentimes there isn't anyone in a person's life to take on that role. You have that opportunity, right now, to be that person.