Question

What is the rule about splitting infinitives? How would you rewrite the quote above based on this rule?

Answer 1

The rule about splitting infinitives is that it should be avoided. If you can avoid inserting a word between TO and the verb, you should. Split infinitives are grammatically incorrect which is why you shouldn't use them.
So, to correct this quote, you should do the following:
"To go boldly where no one has gone before."

Answer 2

"to go boldly where no one has gone before."