Ask Slashdot: How Safe, Really, Is Paying For Things Online?

An anonymous reader writes:
Due to the rash of intrusions into electronic payment systems lately, I've decided to go back to paying cash for everyday purchases, groceries, fuel, and anything else I pay for in person (which also has the positive effect of making balacing my checkbook every month that much easier). The question I have is: For the monthly bills it's just not practical to pay in person (utilities, for instance), how safe are those?
Five minutes of research is telling me that mailing paper checks isn't any more secure than online electronic payments and in fact may be even less secure, but short of literally showing up at the electric company, phone company, ISP, and so on, and paying them cash in person, I can't see any other way to pay them. So how safe is it right now, honestly?
I'm always interested in how Slashdot readers secure their own personal finances -- but how high is the danger that a remote malefactor will hijack and then drain your bank account? Leave your best answers in the comments. How safe, really, is paying for things online?