Martin Luther King, Jr., employs metaphors when he describes the "storms of persecution" and the "winds of police brutality." In using these metaphors, comparing the persecution and brutality endured by African Americans to forces of nature, he gives them weight and draws attention to how significant they are. These are not easy things to overcome; no, they are violent and trying and can hold one back or blow one off course. If you consider the winds that come with storms, they are often particularly brutal and powerful, and so it would make sense that King is still talking about storms when he refers to the winds of police brutality. Again, I believe he wants to draw attention to how significant the problems facing African Americans are, and by comparing the injustice they endure to elemental forces, he achieves this.

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.