When approaching your downed teammate Switch to your Melee Weapon and hold down Right Mouse Button this will cause you to block, and while blocking you hold E to revive your teammate.

It is a very basic thing but so far the game doesn't teach you about it.

With this trick you can take greater risks to revive someone and won't have to fully clear the area of enemies before being able to revive them.

Not a whole lot of people know about this because this was actually a bug in Vermintide 1, but the developers were merciful enough to leave it in the game.