Can you touch the tubes?

I believe this train of thought comes from warnings on head light bulbs in the auto industry.

Quote: When changing the headlight, highbeam or foglight bulb on your vehicle, it is of paramount importance that you not touch the glass part of the bulb with your bare fingers. Oils on you hands will cause the section of the bulb that you touched to get much hotter than other areas of the bulb and will lead to the bulb burning out much quicker. If you do happen to touch the glass part of the bulb, take a solvent like rubbing alcohol and clean the globe thoroughly before installing it in your vehicle. This will remove the oil and the unwanted hotspots.

Wash your hands 1st will minimize oil and give you a better grip. If you can, avoid touching the identifying markings on the glass. If you do, lack of oil will be beneficial.

Whether base or not, use the lower hand for the lifting force, and the other hand on the top, applying any needed minimal movement to help the pins slide up.

The bases are glued on, to give you something to grab/lift. they are no part of the vacuum seal of the glass. The glass may also be broken, a separate issue. If a base becomes loose or partially loose, the tube may be fine. IOW, you still have a working unit. Then you are faced with the decision: buy a pair or quad pair of matched tubes, and keep the loose base as a spare if needed, so you are not without music.

btw, tube testers usually have a few size tube pin straighteners.

For those of you who would still rather wear gloves to pull tubes, these are perfect. A pair came with my PrimaLuna amp. They have gripper 'dots' on the thumb and fingers for a nice positive grip on the tube. I use them even though I agree with most responders that it isn't necessary for tube protection. I just like having a secure grip on the tube when changing one.

If a base is loose, even completely free of the glass envelope, as long as the wires are not cut, you can glue the base and glass together.  I use high temperature epoxy, but I have no idea if that is the best glue to use.

It is not that common for the base to come loose unless a tube is really old.  I run 70-80 year old tubes so I am careful.  Some not so old Western Electric 300B reissues had weak glue joints so care is needed.

I have never heard of anyone squeezing or pinching a tube so hard as to shatter the glass, but I suppose that is possible.  Aside from dropping a tube, the most common accident is inserting a tube incorrectly so that the pins are in the wrong holes.  That can happen if the key on the central post, or the post itself is broken off so thar a tube can be inserted even when incorrectly oriented.  Some 4-pin tubes, like 300Bs, don’t even have a key and rely on two of the pins being fatter; unfortunately, they are not fat enough that they cannot be shoved into the small hole of most tube bases so that a horrible accident is possible. 

Touching tube with bare hands is fine and no danger to health. Many folks do that.