Here's a pair of EL34's that came out of a very unhealthy Traynor YVM-1. Take a close look and compare the bases:

The black cylinder in the center of one of the tube bases is broken off and the bottom of the glass envelope is poking through the bass. This central post is commonly called the key, guidepost or locator pin. It's a bit easier to see here:

So is there anything wrong with using a tube with that black center piece broken off?
Well, yes and no. It doesn't have any electrical function, but you do need to be careful. To explain why I'll start with the bottom of the unbroken tube:

You may be able to make out the numbers labeling the pins, You can also see that the center piece is not a perfect cylinder, it has a small protrusion between pins 1 and 8. I've diagramed both in red here in case it's not completely clear from the photo:

That little protrusion is meant to lock into the groove in the tube socket. With it in place the tube can only plug in in one orientation.

Someone must have tried to force a tube into this socket in the wrong orientation. You can see a chip in the socket made by the tubes key as it was forced in:

My guess is that that is what broke the key off of this tube:

With the key broken off you don't have to fiddle around and align the tube. I can fit it in 8 different ways. That certainly makes things easier. Let's look at the connections made when the tube is inserted. Here's the same socket rotated 180 degrees so the key slot point down:

Now I've overlayed the schematic diagram for an EL34 to show how it's internal elements are connected to those pins.* The black block at the bottom of the diagram indicates the key. I've lined that up with the groove in the socket to show a properly inserted tube. I've also labeled the socket connection with ballpark voltages you would expect to find on those sockets.

Now imagine that the tube is inserted so that the key in the diagram lines up with the chip made in the socket when the tube was forced in. The voltages present on the sockets stay the same the tube elements that those voltages are connected to would be different. I've rotated the tube diagram below to illustrate the point:

The most obvious problem here is that the 445 volts on socket #4 is now connected to pin #7 on the tube.
Inside the tube pin #7 is pin # 2 by the heater element. That effectively connects the 445 volts on socket #4 directly to socket #2. In the amplifier circuit socket #2 is connected to ground through the heater transformer.
This ends up giving the 445 volts a very low resistance path to ground, which translates into a huge amount of current:

Grounding the screen voltage should have blown the fuse but apparently it didn't. It could be that the chip in the socket was just the first attempt and once the key was snapped off the tube was plugged in again in yet a another orientation (once that key is there's only a 1 in 8 chance you'll get it right by guessing). If you do this and you're real lucky the fuse will blow before you cause major damage. This amp wasn't lucky. One side of the output transformer blew. The open transformer winding caused the arcing between pins 2 and 3 on the socket on the other side:

In total the output transformer, the output tubes, screen resitors and tube sockets all needed to be replaced. Not cheap!
So can you still use a tube with the key broken off? Yes, sure. It'll still work if it's aligned right. Just keep in mind the steep price you could pay if you are off by a pin or two, you may not get a second chance.
* You may have noticed that the numbers on the bottom of the tube run clockwise while the numbers on the base and the diagram run counterclockwise. Tube diagrams assume that you are looking at the top of the tube socket. Usually the tube diagram will match the tube base but since we're looking at the bottom of the tube socket in this I've flipped the tube diagram to match the view. Doing that causes the numbers to progress in the reverse direction.

