The president that was known for the statement “Speak softy and carry a big stick” is Theodore Roosevelt.
“Speak softy and carry a big stick” is part of Roosevelt's foreign policy and the Big Stick ideology, which supports the idea of carrying out peaceful negotiations but also being ready to resort to violent methods in case things go wrong, that is, being ready to raise weapons if negotiations do not go as planned. The statement "Speak softy and carry a big stick” was first used by Roosevelt in a letter written to Henry L. Sprague in 1900.