I think the risk of theft, loss etc. far outweighs the benefit.

He will have a mobile phone, a physical credit card, likely further virtual cards through an app etc. And why CHF if the USD is accepted so much more widely?

It depends on the country, but I'd take something like $20 or $50 at most. It's enough for a few phone calls (in case you iPhone isn't working or something), it's enough to pay for a quick cab ride, train ticket, or some basic meal/drink.

EDIT: I rarely carry around much more than $50 when travelling around the world even if I've got my wallet on me. It can be a bit more in countries in which ATM fees are high or ATMs are hard to come by but $50 is a generous average number for me. I even travel without any kind of cash on me in certain countries such as Poland or the UK.