I'll try to accumulate a case here, over time. I think I'll need this to convince a lot of people to get on board with some of my ideas.
In General, it's Useful for Everything
See also The Value of People.
Just like money capital is basically useful for most things. Except that money can't buy everything. Social Capital theoretically can buy more things than money can, I think.
Really, humans are the only thing in the universe which can understand all of your questions. Anything you need to ask for, whether information or material goods or help completing a task, and so on.
Not everyone will though, even if you ask. The more people who will for you for non money-capital reasons, the more "social capital" you have. Something like that.
Specific Uses
Career, employment
It's well known that most jobs are filled by personal referral, not random people applying on the internet.
I'd also speculate that knowing people is kind of like a division of part of the interviewer's labor. Part of the interview is already done. If the person who "conducted the partial interview" is trusted, at least. Same thing with the references people put on their resume's (knowledge akin to an interview is gained, if that communication is trusted).
Political action
Few people take any political action alone...
Social connections can help solve Coordination Problems.
It takes a village to raise a child
Pretty self-explanatory, though this point can be expanded...
Health
People who have good social ties have healthier, longer lives. And probably more fun.
Catching up to our Competitors
Most non-humanists have a lot of social capital already. For example, in churches. The Catholic church is kind of the biggest example of this. It's huge and long lasting, but it is (at best) very slow to progress over time, and is in no way scientific.
Others (who may or may not be very religious) have lots of regular capital (money, property, etc.).
More examples:
Studies and Research that Demonstrate the Importance
Saw one about getting people to work towards shared political goal type thing. Bubble groups of people that don't conect to each other may fail to get things done as a whole, perhaps partly because not all the bubbles are on the same page (or they don't know if they are or not), and because outsiders don't compel people's actions as well as insiders.