Management is thinking of giving me more responsibilities. I am open to this. When exactly should I ask for a pay raise? When it is confirmed that I will be getting new responsibilities (e.g. I am scheduled to be trained on them)? Or after the fact, for example "hey, now that I'm doing more I would like a raise"? The downside I see to the first option is management may just find someone else who didn't ask for a pay raise.
Have this discussion with management when you talk about your new responsibilities. Could be a real step up, could be a lateral move or could be just normal growth of your day-to-day work.
Have a career discussion with your manager, determine what the next step up for you is, what's required to take the step and how the new responsibility fit in there.
Do your new responsibilities warrant a raise? Answer that question first.
If the answer is yes, then ask for a raise to coincide with these new responsibilities.
As you've stated, the downside is that they could find someone else to give these responsibilities to. That's a risk you'll need to take if you want the raise.
The upside for the company is that you won't ask for a raise, and that nobody will ever ask for a raise, and that they can continue to pay people the same while giving them ever increasing responsibilities and workload.
My point being, don't worry about whether or not asking for a raise is risky. If the responsibilities warrant a raise, then ask for it.
When exactly should I ask for a pay raise?
You indicate that currently they are only "thinking about" giving you more responsibilities. So don't ask now.
When you are formally told that you are promoted/more responsible, see if it comes with more money. If you don't hear that it does, just ask "Does this change come with a salary increase?" Be ready for how you intend to react in either the Yes or No case.
Some companies increase your wages at the same time you get promoted. Others increase your wages later.