If you can directly print from your computer/phone to a printer (via USB/network) then you're probably fine. Similarly, using a photocopier just to make simple copies is probably fine too.
I definitely would not trust a computer in the business center, not even to check in for a flight.

There is a good argument to also NOT trust scanners and printers--they're just more restrictive computers. They do run software and they're likely connected to a network, so yeah...they could totally steal information. At the same time, it's just not that likely someone is trying to hack probably seldom-used a printer or copier in a random hotel and so I'd probably use one if I needed to.

I said 'probably' a lot because you can't actually make guarantees about a device's security if you haven't audited it. I would definitely be cautious about even printing something like account tokens / backup security codes, but I probably wouldn't think too hard about printing a bank statement or other moderately sensitive data.