Anecdotes and testimonials: "One swallow does not a summer make". The natural medicine industry and vaccine scaremongering is drenched in anecdotes and testimonials, such as "Mother X gave her son the vaccine, and he developed symptoms of autism", or "Child Y has never received a vaccine, and is perfectly healthy". These anecdotes are often backed up with dubious logic like "are you saying a mother doesn't know her child?". Problem is, whether the anecdotes are true is not related to the question of whether vaccines are safe, as each anecdote, even if true, is just one story. For every anecdote a vaccine "sceptic" brings to the table, I can counter him or her with ten anecdotes about how a child got his or her vaccine and was perfectly fine. Really, I shouldn't have to say this, but to determine whether something is too dangerous to be worth taking, you have to look at the big picture.
Here's a statistic to really put this into perspective: A whooping 30 000 people are killed annually by seat belts and air bags. This is a shocking number, but the reason you won't find huge many lobbies today fighting seat belt laws is that we know for a fact that 30 000 is a tiny number compared to the number of people whose lives are saved on a daily basis by seat belts and air bags. I could list horror stories all day about how this little kid or that pregnant woman was maimed or killed by air bags or seat belts, but this doesn't change that you're far more likely to die in an auto accident if you don't wear a seat belt. We know for a fact that seat belts and air bags save far more lives than they take, so we're not swayed by scare-mongering like "this little nine-year old girl on her way home from handball practice got into a car accident and died because she wore a seat belt". For the same reason, no one should pay attention to anecdotes, no matter how many, about the dangers of anything else, from elevators to vaccines to eating of bread.
Unrelated circumstances: whether a product is "natural", made by a big company, or profitable, or contains ingredient x, has absolutely zilch to do with the discussion. If you want to determine if something is good for you, you ask if it's good for you. Circumventing that question by throwing mud like "Offit made money on the vaccine!" or "the vaccine is not natural!" does not answer anything and merely loses you credibility. This kind of argument is known as a "red herring", from the old belief that dragging a red herring across your trail could throw a bloodhound off your scent. They're meant to derail the discussion and give one side an advantage is does not deserve. Recognize red herrings when you see them, and stay clear.
Unfounded and unrelated personal attacks: Person A makes an argument against your position, therefore he is a "sheep" who blindly trusts Big Pharma. Person B got struck off, therefore there is a Big Pharma conspiracy to silence dissent. Person C makes an argument you dislike, therefore he or she is a Big Pharma shill. Vaccines made Person D rich, therefore they are unsafe. The list goes on. Personal attacks need to be backed up with facts, and they need to be directly on topic, that is, answering the question at hand, which in this case is "should we get vaccinated". Saying that Dr. Offit got rich off of the vaccine could mean there's an industry secretly pulling threads to silence dissent, but it could just as well indicate that the vaccine is a successful product that does what it's intended to do. Either way, it does not answer the question of whether or not you should vaccinate your child, so it's irrelevant to the discussion.
When confronted with an idea, a very good way to see if it's worth paying attention to is to check how rationally it's being promoted. The fight against vaccines is fought largelly by means of red herrings, personal attacks, conspiracy theories and scare-mongering, and very little in the way of actual empirical studies. What we do have of such studies demonstrate overwhelmingly that vaccines do work, that they are a huge benefit to humanity, and that the risks are far outweighed by the positive results.