Voltage and current are just different ways to measure the same phenomenon. Both happen at the same time. The reason that we talk about the hazard of voltage rather than current is that most sources of electrical energy in our world have a particular voltage (voltage source) rather than a particular current (current source).

Also, very low currents can be hazardous but skin resistance is relatively high. Most of the voltage sources we are exposed to are fairly low. There is no point in putting a warning about the 1 Amp of current a power adapter can produce if that adapter is limited to producing 5 Volts. In a world where all power sources were producing thousands of volts the opposite would be true, an example warning would be "Danger, 200 mA".

In general we have to worry about the exception, not the rule.