I use latex when handling circulated coins and cotton gloves for proofs. That's how I do it and it seems to work out best that way.
I use lintless cotton gloves. But they DO make picking up coins a challenge. My hands turn into prunes if I use latex. And some people are allergic to latex. But for me using gloves is basically moot since all of the coins that I would want to handle carefully are in slabs.
I would avoid using gloves. They reduce your dexterity and thus you are at greater risk of dropping the coin. If you properly handle the coin, no gloves are needed. Also, those blue gloves look like nitrile gloves to me...not latex.
I should have been more specific. I don't use latex gloves for handling just a few circulation coins. In that case I use bare hands also. I use the gloves when handling many coins in one sitting, such as in roll hunting. For proofs I always handle them with cotton gloves when outside of a case (aka raw).
The use of gloves, of any kind, to handle coins is probably the worst advice that could ever be given.
I can understand wearing gloves if you are roll searching...to keep your hands clean as money is quite filthy. However, I think wearing gloves to "protect" the coin actually puts the coin at greater risk of damage due to the loss of dexterity.
Loss of Dexterity. I wear gloves all day long...things are far harder to hang on to when you have a layer of material between them and an object. You risk dropping the coin.
I understand the loss of dexterity/grip thing, but I'm not handling the coins standing up or from a great height. I'm usually sitting at my desk or a square table I have that has a padded top. When at my desk I usually have a soft cloth/rag below my hands.
Cotton will leave scratches on the surfaces of coins. Just use a cotton squab on a unc coin, and you will see hair like scratches on the surfaces with a loop. I just use my fingers, and be careful handling any coin. I used a cotton squab to clean a unc coin that had some grease on it. After this I noticed the scratches, and do not use them now. To remove the grease, I would dip the coin in acetone.
I never handle a proof (which is when I would be wearing cotton gloves) by the faces. Therefore it's not possible to scratch the surface with the gloves.
It's still better to not drop them. The gloves pose a greater risk than your fingers. There is no benefit to them and there is risk.
On a similar note, I teach an elementary college chemistry lab and students often want to wear gloves. I discourage this since it does cause you to lose dexterity and there are few chemicals we handle which you could not simply rinse off if you get them on you.
I used cotton gloves 20 years ago to grab hands full of cents from mint bags. They were good for that. I'll never use gloves again.
I don't see anything wrong with it...in fact, they are doing it completely correct. They are holding the coin by the rim without using any kind of glove. This is the safest way to handle a coin.