The appeal here is that the course is free, ergo, _anyone_ with a basic internet connection can take it.
If you require payments, you then have credit card companies and Paypal restricting who (i.e. which countries) gets access to it.
Income from donations isn't guaranteed, and thus provides a weak incentive for universities to provide these kinds of courses. I'd also wager that donations would fizzle out as this sort of thing became more commonplace.
US Government grants would provide guaranteed incentives for American universities to offer these kinds of courses, which would educate your populace and bolster the appeal of your higher education institutions to potential overseas students. The latter point would lead to many knock-on economic benefits (e.g. immigrant students deciding to stay in the US).
(oh, but rabble rabble gubment rabble bad rabble rabble because this is HN where libertarian capitalists roam free)