First, you need to know that a lot of people don't like it when you make a remark about their conduct still because they feel attracted. I know it's true for my mother and also for every person I know that isn't really confident in their driving skills.
I don't know how people with a lot of confidence in their driving skills will react, however, even with a lot of confidence, being criticized never feels nice.
So, when you will ask your coworker to drive more safely, you will need to be extra careful.
In those kinds of situation, one thing I learned is to make everything about you.
So, when I need a favor from someone and I'm afraid they will take it badly, here is what I usually do:
- Start by saying:
I'm sorry
This signal to the other person that you are going to tell them something and that they probably won't be happy about it. It's important because it gives them time to mentally prepare for what will come next.
- Follow by saying something like:
I'm kind of an anxious person
Here, you are making thinks about you. So, when you will ask them to drive safer, they will know/believe that it's because you are an anxious person (more than average) and not because their driving is terrible (even if it's terrible indeed).
- Finish by asking the favor:
Could you not look at your phone when driving me? It makes me very anxious.
Several things to say here: I said "not look at your phone" instead of "driving more safely". "Driving safely" is kind of vague and people don't always don't know what you mean by that. Also, you might not have the same definition of "safe" as the other person which mean, even if, by their standards, they can be driving more safely, you might still not be satisfied by the result. So, telling them about something concrete they could do will likely lead to better results (this is the case for me when I ask someone to improve their question/answer).
The other thing to note here is the second sentence. It some modified version of the "I" statement technic. Stating why you want someone to do something help them understand the actual issue and they become (in my experience) more likely to do what you asked.
##A little bit more about the "I" statement technic
This technic work as following:
When you do X,
It makes me feel Y,
Could you do Z instead?
You can learn more about it here or in a lot of other place in the internet.