How to get the red key in Doom 64 level 7, Research Lab:

Once you get the yellow key, find a yellow-keyed door and
go through.
Enter the room ahead, where you'll find 3 stairways: One
central stairway leading up, and two more to either side
leading down.
Take the left staircase down, and you'll come to one end of
a slime-floored tunnel.
Walk along the narrow ledges on the side of the tunnel to
avoid being burned by the slime.  Turn left and look for a
barrier of three vertical bars.  These will rise up as you
approach.
Step past the bars and turn left.  Look for a small alcove
in the wall to the left, and pick up the radiation shield.
Now continue down that tunnel until you come to a dead end,
where you'll see a switch.
Activate it, then turn and follow the tunnel all the way to
its opposite end, where you'll find another switch.
Activate this switch, then head back to the juncture where
the 3-bar barrier was.  Go through the barrier and return
back up the steps to the "3 stairways" room.
Go up the central stairway to the room ahead, then follow
the next set of steps counterclockwise up along the walls
of the room.
At the top of the steps, you'll see a switch ahead of you.
Activate it, then go back down into the "3 stairways" room.
When you get to the bottom of the central staircase, you'll
see the two yellow-keyed doors, just ahead and to your left
and right.
Go through the right-hand door, then proceed to the first
door on your left.  Open it and go through to find the red
key just off to your left.