What power do I have as an E3 to fix this E2?

The thing is most enlisted see thing are easier then basic and they just asume that it ok to talk on a phone and such, It best to talk to him/her in manner and ask if they use they phone on break, But best to let your NCO knows, We are all a team, but it more right to let nco handle it, we don't tell them to do push up or punish them just because he's a pricate and you a specialist.

As several people have already said it depends on your unit and amount of delegated authority. The one thing I can tell you that hasn't been suggested is use YOUR phone against him: take pics and/or video and pass them onto your or his NCO's.
Either way, you're doing the right thing by trying to do something about him.

Depends on your unit, but more importantly. What was fine when I was a boot (ie hazing in any way they could imagine) is NJP-able this year.

If your command wants to coddle your boots, unless you don't care about getting in serious trouble, there's nothing you can do about it.

One of my boots didn't have his gear packed after his team leader had told him to for inspection. He told him he'd be back at x-hours to reinspect. When he came back, the boot was at the movies because his mom said, "you should do whatever you want to. The weekend is your time."

He got away with it. Something I would have gotten put on superman firewatch for and generally fucked with, he had no repercussions.

What they don't understand is that they're making these boots weak and basically sentencing them to death for the next conflict.

Whatever. I won't be here.

There are several forms of counseling.

Firstly you should never look at it as "fixing" someone. Generally it is not the lower rank service members problem more than the peer's and command structure's lack of proper training and remediation. Find out how you as a peer, even as a friend, can help that individual through the issue. Turn the task into something fun and rewarding if you have to. You don't always have to think inside the box. You will better yourself and those around you if you break out of the OODA loop. Asymmetric Warfare and Thinking, It’s not just for the “Other Guys.”

Try to train together, keeping in mind while you train of the OODA loop. The more you train, the more you practice, you can start removing Orient and Decide leaving only Observe and Act.

OODA Loop:
Observe The initial stimulus triggers a response.

For example, you are in a dark room and a TV set goes on unexpectedly. It gets your attention and you look in its’ direction – whats that?

Orient Identification of Stimulus

Oh, that’s the T.V. and that’s the Seinfield show.

Decide Decision to Act / Choice of Action

I’ve seen that episode – I’m going to change channels.

Action The physical act of carrying out the decision. You reach out and change the channel to another show.

Unfortunately you won't have much authority to do anything unless you are placed as his team leader or some position that merits that authority. At your level you can warn him of the regulations unit level and army wide of phone use and soldier requirements (Maintaining competencies in basic tasks) and advise him that violating these obligations will force you to inform senior leadership.

Make him care or make some above him make him care.

Not your problem, do what you need to do for yourself. If your leadership is already aware pf the problem like you say then it is firmly on their shoulders. Don't let the kid drag you down. You aren't getting paid to make those decisions or try to fix people.

See what the your Company SOP is on cell phone. If the SOP are against them inform the SM. After that either you or your sergeant take his phone away from him or if you are in charge of him, put on all the shit detail you can-the dirtier the better. If he hasn't learn yet, take it to the next step.
Now me personally, I would told him once to put it up or I would break it. I was known to be an ass at time lol

I would suggest you talk with your NCO's about the problem. Anything you do on your own can and probably will make things worse.