Only three kinds of crates in Command & Conquer: Red Alert 3 and Command & Conquer: Red Alert 3: Uprising: Credits, health and veterancy. And these cube-shaped items clearly label their contents by different symbols on their sides. Commander can consider to pick up or not by his own needs.
Like Command & Conquer: Red Alert 2, health and veterancy crates have field effects, so better send units in tight formation to pick them up.
In campaigns, crates are pre-scripted on battlefields. In skirmish and multiplayer mode, crates are a kind of "scenario condition" that can be switched on/off before game starts. If switched on, crates randomly appears on battlefield and, if not being picked up, will vanish after a certain time.
To pick up a crate, simply order a ground unit (if it's on water, order a ship or submarine or amphibious vehicle or anything can move on/beneath water) to move to its position, mind many crates have field effects.