Valerie Jarrett's blog post is frankly disturbing...
"Every day, we lose twenty-two veterans to suicide." That's bad enough, but what Valerie Jarrett would not be in a position to know is that, if veterans (or anyone else) call tax-funded "crisis lines," the lack of practical help they encounter may make them more not less likely to commit suicide.
The insurance industry has the "volunteer counsellors" in such a panic about homicidal stalkers that, whatever the facts of the situation and the needs of a caller may be, all the "crisis workers" are supposed to do is push anyone who seems at all emotionally affected by a situation toward what is broadly and optimistically classified as "professional help." This "help" used to include support groups and pastoral counselling, but these days, in a frantic effort to supply some sort of proof that someone out there might actually be "helped," unhappy people are pushed toward people who dispense feel-good pills. And unfortunately, although popular antidepressants give most people enough of a mild "high" to get them hooked, for one out of every ten or twenty users the predictable side effects include an appallingly consistent pattern of physical pain, pseudomemories of horrible physical abuse, urges to seek revenge and/or protect others from similar abuse, and an urge to commit suicide after killing as many other people as the patient believes s/he can.
Caring about people, appreciating what they've done, convincing older people that their lives aren't over, are not tasks that can be delegated to specialists. These things must be done in very specific individual ways. If they're not done in the right way by the right person, for many of us, they can't be done at all.
Veterans are only somewhat more likely than other middle-aged people to have suffered brain traumas that produce emotional disorders (and/or long-term pain and/or disability due to nerve damage). All middle-aged people are, however, likely to notice that in these crowded United States a lot of people wish we'd just go ahead and die. That's a fact. If you prefer that a particular middle-aged or old American stay alive, the thing to do is not deny the fact, but affirm that you want this person to be alive as much as, or more than, others may want him or her...well...out of their way.
Actually, although your words and your feelings may be all very well, they may not do much for the person you're trying to encourage. Your emotional connection with this person depends on a reasonable understanding of your relationship with the person. If you're not a close relative or Significant Other or long-term friend, it's better to be a good student or customer than it is to try to pretend that your friendship means anything to the person.
So how should you thank a veteran (or some other older person)? How would I know? It depends on who that person is, what that person does, what's important to that person. If the person talks to you, s/he will probably give you better clues than a general article could.
I can suggest that you show appreciation of what any older person has already done. Here are some ways our crowded United States' culture trains us to do the opposite of that: suggesting, much less (God help us) requiring, that the person have any form of contact with social workers; ignoring the facts in the person's complaints and trying to fix the feelings, which would probably sort themselves out if the facts were fixed; ignoring the person's achievements and trying to treat an experienced adult the same way we might treat a fourteen-year-old, whether that's referring a disabled veteran to an entry-level job, or calling a retired person "Buddy" or "Sweetie." I can tell you that although I'm neither a veteran nor depressive nor suicidal, as a person over age forty I find any of those things extremely discouraging...but they may not even make the top ten list of your older friend's pet peeves.
What you can do is ask. If the veteran or other discouraged older person you have in mind is your parent, you probably know what s/he would find encouraging, so do it already. If it's someone you don't know so well, you can always say that you're reading an online discussion about veterans and senior citizens and you'd like to find out what more people in that category would consider good ways to show appreciation of what they've done.