Can my employer dock me half a day when I am only a bit late to work?

Did you agree to such docking, in writing? See the labor and employment link from New York State's Department of Labor for details: https://labor.ny.gov/formsdocs/wp/LS605.PDF What is the maximum deduction established by your employer per New York's Wage and Hour Law? Is it at or below the maximum permitted by the Wage and Hour Law? If no union contract protects you, no private fixed term employment contracts protect you, and you do not work for the government you probably should arrive earlier than late, ever, to work (even if you have to deal with slower, hot, summer public transportation issues). Tardiness may be the easiest way to terminate an employee as long as the policy is applied uniformly, across the board. You do realize that "at will" employment, assuming none of the above possible protections apply, means that the employer needs absolutely no reason at all to terminate an employee. Don't give your employer an easy reason when they don't even need one.