Mr. Strunk and Mr. White in The Elements of Style urge writers to avoid starting a sentence with “However.” It’s not a grammatical mistake to start a sentence with However. It’s just that good writers don’t do it. Instead, they start a contrasting sentence with But.
You would start a sentence with However only when it means “in whatever way” or “to whatever extent.” Thus:
However foolish his actions might seem, he is, after all, only a teenager.
If you use however to join two independent clauses, you must end the first clause with a semicolon and put a comma after however. Using a comma before the however is a gross mistake. Thus, the following is incorrect:
She enjoyed the beach, however, she forgot her cooler of beer.
She enjoyed the beach; however, she forgot her cooler of beer.
Example: “But they also knew that [free speech] is always the deadliest enemy of tyranny.” Hugo Black, The Bill of Rights, 35 N.Y.U. L. Rev. 865, 880‑81 (1960).