When an E-4 calls me "hey" or "hey man" knowing that I am an officer, should I bother to correct him?

You are still an Officer and he is Enlisted, whether in uniform or civilian clothes he knows that you are an Officer and must address you as such.

Most fraternization infractions happen out of uniform. Best to professionally nip this in the bud.

a soldier is a soldier 24 x 7 x 365. In or out of uniform, if a person is recognized as an officer due respect should be rendered. As noted by others in this string, failure to correct that lowered the standard.

If you're unsure about correcting an E-4 in Garrison about whether or not he should call you "hey man" you probably shouldn't be an officer. If this decision is paralyzing you, just imagine what combat might be like. Best of luck SIR!

Off duty and out of uniform you are just another guy. If he showed respect to your rank while on duty then what's the problem. If you are both in uniform when he is addressing you as "hey man" then yes this is obviously a problem that needs to be corrected.

Put him at attention and CORRECT him. With emphasis.

Out of uniform. What should happen is lower ranking does not know you are of higher rank or grade. You inform them of thier mistake and then prove you are the rank you claim. I used my ID card. After establishing who you are and who they are, you should get the respect due to your rank. If soldier continues disrespecting your rank then you ask the soldier for his unit. This normally gets the correct results. If not contact his chain of concern depending on how upset you are would determine brigade or battalion CSM. You could also contact his or hers 1SG if you are feeling nice.
This has the affect of making you look very much like you are shoving your rank down someone’s throat. People will hear and treat you differently. It never really bugged me. After 25 years I just stopped caring about appearances. I always reminded them and myself that if someone of higher rank gets treated like this, well it can go very bad and quickly. You shouldn’t feel like you are being a prick. We are soldiers 24 7 in or out of the uniform and some soldiers need to be reminded.

What kind of leader do you want to be known as? An officer or want to be?

I was always taught from basic training to my retirement from a military installation, that if you recognize and officer or senior NCO, show respect by giving the greetings of the day whether in or out of uniform. You may not respect the person, but you must respect the rank. Question. Do you address individual as "soldier" or warrior" off post? Maybe the individual in trying to conceal the fact that you are an officer while off post.

Are you really asking this question as a Captain in the infantry?!?!?