Can you touch the tubes?

Yes you can touch them 

For better grip I use nitrile gloves. 

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.