In a public library children's area context the application of the theory might be a little more difficult to apply to patrons. Regulars might be slightly easier, but, without an extended interview, how can you determine a person's (adult or child) achievement motivation? It would be very difficult and imprecise. But, you can take individual comments as you help someone.

For example, in a reader's advisory context:

"So my mom told me I have to find a book to read." This could be either power (subtext of "I wouldn't be here if she didn't make me, so I'm going to keep the power in this conversation at least and tell you as little as possible since information is power) or affiliation (I have no desire to be here but my mom wants me to do this and I want to make her happy). 
"Ok, what kind of book would you like?"
"Something with dragons." Power again, for the same reason. 
"Do you like to read?" 
"No. I hate reading." Power since he might expect the librarian to like reading and wants to upset her, keeping the power in the conversation..
"Is it easy for you?"

"I mix up all the letters. It's hard to keep them straight." Most likely achievement since our patron won't do something he's has little chance of succeeding at and he has a difficult time admitting that. 
"That's ok. I hated reading when I was your age too. But I really liked graphic novels because the story was in the pictures. Would you like to look at one of those?"

"My mom said it has to be a book to read..." Affiliation since he wants his mother to accept his selection. Power because, despite his interest he doesn't want to agree because it hands power over. 

In this short exchange we might think that this patron has a high personal power need. When the patron comes back we can take that into account in our interactions. Perhaps not approach him unless necessary, giving him the control of when we can help him. Don't tell him what he should or shouldn't read, that he has to read, or any other things he should or shouldn't do unless it's a safety or library rule concern. 

If we had identified an affiliation need, then we might always smile and ask how the patron is, if there anything we can help him with. With and achievement need we could ask how the patron made out with the last book tailoring our next recommendation so it's not too easy or too difficult.

I'd tell my staff though that it's important to take into account that at certain ages one need might seem more prevalent in everyone, especially in short conversations. That doesn't mean we shouldn't still try and understand and meet that need, but that we should keep in mind that the need might change. If our example patron was a teenager, it's likely the power need that he expressed was a normal teenage reaction to a parent interfering in his life. In additional library visits by his choice, he might reveal himself as achievement motivated. If that was the case then an observant librarian who recognized that would be better prepared to help him overcome his probable dyslexia.