I rarely ask how old someone is, and I have no problem telling people how old I am.
I apparently look much younger than I am, and have actually had to dig out my drivers license to prove that I am <insert age> [whenever they asked me how old I was, last time was about 3 years ago]
I am currently 49.
I don’t generally ask others - it’s considered rude. But I have no problem being asked. I’ve got no issues with my age.
When I ask someone his/her age it’s not that the answer is “important,” it’s just one more piece of information about him/her. I don’t go around asking everyone I meet how old they are, but sometimes I’ll ask if someone says something that implies they are significantly younger or older than I thought. It usually comes up during conversations that include pop culture references. Sometimes I wish my age were as obvious as my skin color, sex, height, etc.
I used to think that. I’m 39, and most people think I’m in my late 20s…it’s not the worst problem to have, but it never really stops getting old. Sorry. 
I ask people who are likely to answer by holding up the right number of pudgy litttle kid fingers.
If I’m talking with someone and become curious I might ask “when did you graduate from high school?” or some other way that relates to the conversation.
I don’t ask other people, if just because it rarely comes up. Well, no, that’s not quite right. I don’t ask other ADULTS. I’ll certainly ask kids, because sometimes it’s relevant for my work, and most of them are all too eager to tell me that they’re 6 years old.
As for being asked, I don’t mind. I used to get the “you look 16!” bit (I’m 23), but since I’ve gotten a super short haircut, I probably look my age now.
Wow. I thought the only people who were offended by talking about age were old ladies in the 1950s. 
It doesn’t come up that much, but I freely ask and answer if it does. (And it has more lately since I remarried a woman 9 years my junior.) I’m 44 but people guess mid 30s, and I got carded for alcohol from time to time. (15 days sober, so I use the past tense). Seeing that it offends some people, maybe I should be more careful. That said I would never ask to insinuate incompetence or anything. It’s just a topic that helps you understand what the other person’s experiences are likely to have been. Like the example above about getting cultural references from the 1970s.
This. It usually occurs to me when I’m trying to get a “pop culture” age, so the conversation makes sense.
I often find that referring to “Convoy” with the young law clerks gets me nothing but blank stares.