The word "winds" is a great stylistic choice by Martin Luther King, Jr. for several reasons.
Yes, it could mean storm. Winds are a pretty vague term, but in general, wind is associated with a buffeting force that can push people around. In this case, it could push Americans who were black away from their goals or push them around. An important accompanying word is "staggered" - this is a common verb that provokes the image of a person pushing against a force, like wind. What is important about this word choice is that a person who is staggering has not fallen, and is continuing to persevere.
Another connotation of "wind" as the opposing force: Wind, unlike policemen, is actually pretty insubstantial. Wind has no mass in itself - it is simply air rushing from one place to the next. By comparing policemen to wind, MLK Jr. acknowledges that they are prolonging the goals of the Civil Rights Movement, but also noting that, like wind, they are insubstantial and will never be able to stop it entirely.