Just because a power supply is rated (ie. capable of) a higher current does not mean that this is what the 'sink' (ie. Raspi) will take. This is different to voltage, where excess voltage can indeed damage a circuit that doesn't have sufficient protection (eg. voltage regulators).
Think of it with a battery. Most batteries are actually capable of quite high currents (albeit with a shortened life). However, something like a flashlight/torch will only take a current according to its design - and this is typically a lot less than what the batteries are capable of.
After saying all that, you should try to avoid a power supply with an extremely excessive current rating because such a set-up will prove to be inefficient in operation and uneconomic (relatively expensive to buy - although it might be one you have spare).
So 1 amp is fine and probably gives a healthy margin assuming you don't have many peripherals powered from it, but 5 amps would be excessive.