Tom Goldman said:
the response by many in the audience wasn't to help Nye, but to update their Twitter accounts with news of what had happened
That's actually the proper type of response.
Cluttering around somebody that just fell down is one of the stupidest things you can do... it only takes one or maybe very few people to help, and a crowd gathered to gawk hinders assistance more than it could possibly help.
If nobody else seems to want to help or if you have good enough reasons to believe there is no staff doctor that would need free passage to the stage... then yeah, you get up and go help (if you think you CAN help), or ask around if anybody IS a doctor.