What's the appropriate verb to describe the action the librarian does when you borrow a book at the library, something like hand or give?
You borrow a book at the library. The librarian issues a book.
2a : to put forth or distribute usually officially
Examples of ISSUE
Each employee will be issued an identification card.
The Post Office will issue a new first-class stamp.
The company plans to raise money by issuing more stock.
The bank will be issuing a new credit card.
the bank's newly issued credit card.
The king issued a decree forbidding all protests.
A severe storm warning has been issued.
The police have issued a warrant for her arrest.
[Edit-1: Per @Mustafa & @sidewaysmilk]
Lend is the usual alternative, esp. in the US.
From my decade of experience as a librarian, I would say that in most libraries, the librarian does not check out books. That job is usually done by a library clerk... But maybe I'm being facetious.
In the US, we often use 'loan'. The books is on loan. Request an inter-library loan... The library loaned the book to the patron.
Anyway, the librarian (or whoever) does not loan the book. The librarian doesn't own the book. The Library loan/lends the book. The librarian checks the book out.
Or 'The library clerk scanned the barcode on the inside front cover of the book, tore off the return slip from the receipt printer, demagnified the security strip and slid the books back across the counter.' Because, you know, that's actually what happens.
Not issue. Issue has a connotation of the thing(s) being issued being newly created for that purpose. Similarly, reissue usually means "release more copies of the the same thing", not e.g. "reship".
I would use check out. Check out is ambiguous in that both the patron and the librarian can be said to check out the book, but nevertheless is the verb most commonly used by librarians, AFAICT. In this context, it would probably be used as "check out [the book] to [the patron]".
I'm in the USA.
Library books are borrowed and lent in the US, parking tickets are issued; indeed issue has an immediate negative connotation (politicians tangle with issues), and libraries are a nonjudgemental, demos-affirming institution. Library cards are issued, books are borrowed - issuance is more official than lend, and lacks the one quality of borrowing - the explicit right of return.
Government Issue (G.I.) is a term which describes the assignment by the military of personal effects, clothing, weapons, tools etc to the individual without relinquishing ownership.
If it is the process of organisig for a person to borrow a book, issue is the right answer in the UK as per @Pitarau. Then the book is "on loan" to the borrower.
The action whereby the potential borrower identifies, has issued, and then takes the book away, eventually returning it is a "loan". This is a far wider process.
When a borrower has some books they wish to borrow, they may ask if they can take them out. At which point, they will hopefully be issued, assuming that the borrower is allowed to do this.
It is all a bit of a labyrinth.
I would agree with the "Issue" suggestions if the context is to describe from a "process view" the act of a hypothetical librarian fulfilling her role in providing books to members.
However, I would also say that if the goal is to describe a librarian as a character in a novel or story, or as a real person in an article, then the setting and the personality of the librarian come into play. I can only imagine the most severe, stoic, librarian "issuing" a book while I could see, again based on personality and setting, the book being "slid across the counter", "tossed", "dropped into waiting hands", "pressed into hands" etc.
Update: On further thought I am thinking my answer belongs more on the Writers site. ;-)