This first question that might be asked is: what exactly is a den. The word "den" is not used as much as it used to be but turning the basement into a den is one of the better basement ideas you can come up with.
There is no one definition of the word den but it is usually thought of as a room where people go for a variety of purposes. When offices where less common in houses (now many people have a "basement office"), the den was often used as an office. That was the best place to put a desk and some bookshelves along with maybe a fax machine. Now so many people have offices in their homes that they are rarely referred to as dens.
A den is also a room that is used as a gathering place. The den can be a television room and a place where a family goes to just hang out. Not all homes have a family room for this and so the den is the place they go.
A den can also be used as a quite place to go for reading and to get away from the noisier parts of the house. If you want a smoking room or a cigar room, a den would be a great place for that purpose.
Those are some basement den ideas if you have a basement and don't know what to do with it. Basically a den can be whatever you want it to be...sort of a multipurpose if you so choose. Turning your basement into a den is great idea if you need extra space for your family members to spend time and hang out.