How do I build trust while teaching responsibility?

On a side note in a slightly humorous way and kind of from past experience, as a 1SG I always told the troops to call me if they even sniffed a beer before getting behind the wheel. If you have a young troop that is lets say, in an altered state without a solid plan to get back on post the first favor is a freebee. When you get this troop to the barracks and he sobers up the next duty day you sit him down and help him plan out his actions for future reference with an advisory that his next uber will be with the PSG/1SG and he will be guaranteed to get squared away lickety split. I always gave one freebie for a ride as I would rather loses a few hours sleep than explain to my commander/BN CSM/BNCO and assorted other folks why my troop received a DUI/Died in a car accident or tore up the post commanders front lawn showing off. If I received a 2nd call, lets say I gave them much additional attention 24 x 7.
If you don't nip this behavior in the bud right off the bat it will become a nightly issue until one gets a DUI/Car accident with death or injury. There is being there for your troops and their is being used due to lazy behavior. I once picked up a troop that was a bit tipsy and dropped him off a nice walk from the barracks which I followed him in my car. He never did that again and he was good by the time he arrived at the barracks. Its easier to fix this now then after a big screw up. Sometimes examples have to be made.
Back in the day :) it was the norm for almost an entire platoon to drink into the wee hours and yet still be out in formation "ON TIME" in PT uniform for the 1SG's sober death runs. The smells, sights and levels of drinking were sight to see but everyone made it and they were sober real fast.

It's a balancing act. One COA for you is this:

Go pick them up. Since they are probably drunk, there is no point and having the "talk" on that ride back to the barracks. Once sober and at work, let them know that the ride received is something that you can and will do when needed, but not to abuse that option. Provide them with all available means for alternate ride. Give them the local cab number that has the capability to get on post. Have them download the Uber and/or Lyft app. And watch them download it. In fact, I would say to sit down with your team and have a counseling en mass about this and ensure they ALL have the alternate means of transport in their phones in case you can't be reached.

The fact that they can rely on you for this kind of "emergency" is a good thing. And yes, them getting themselves to work is their responsibility, but it is your responsibility to ensure they are at work. If they aren't at work, guess who the first person the squad leader is gonna turn to? You. And what would you say if they miss work...a search party is sent out and find out something bad happened to them somewhere between the gate and the barracks?

Well not knowing more - pick up individual and get them to work - that simple. In fact thank them.

"Hey man thanks for letting me know the situation on getting to work. Call me anytime with shit like this"

I've had to drag many individuals out bars and out of bed all the way up to BN XO. No it wasn't my job but it kept people out of trouble and from my experience when a platoon/company/BN becomes a problem child for higher ups it usually results in some top down attention that will without a doubt somehow make your life a little less enjoyable.

Your leaders will eventually see that you're looking after your squad which in realty is looking out for your NCOIC and CoC and it will be rewarded.

Just be glad this individual reached out to you for help -