I just acquired a set of 45 year old clubs that have the original, leather grips installed. Considering their age they appear to be in fine shape, but my (bare) hands don’t feel quite secure on them. It is possible that I could get used to them (I haven’t even hit them yet). Nonetheless, if anyone has any tips on how to improve or maintain these grips, I would love to hear them. There is a constaint: I would strongly prefer it if I wouldn’t have to remove the leather from the shafts while cleaning or otherwise caring for them.
There are several leather care products that will work. One is Lexol that is available at most shoe repair stores and sometimes at leather product stores. Lexol also has a cleaner that works well. Saddlery/tack shops may also have good products. Lexol is best used in several light coats rather than one or two heavy coats. Use it often and wipe the grips clean with a damp towel to keep the grime away. Mike
Thank you! It looks like I can pick up both Lexol conditioner and cleaner at Autozone, which is convenient enough! That is where I got my paintfill applicator and chrome-polish as well. 
I have picked up the Lenox products. I will apply them tomorrow.
In the meantime, a question has come up. 
The clubs I am going to treat are 1960’s irons. It seems to me that the grips are much thinner than standard (rubber) grips are these days, especially near the butt-end of the club. Have the grips somehow shrunk, or were grips like that back in the day? If so, is that good, bad, or something else?
I played my ‘new’ set today (1967 W/S X31s). I had not practiced or played with them beforehand. I liked them! The one thing I was worried about was the leather grips. On hole #1 I put on a LH glove, when I normally don’t wear a glove, just in case. On hole #4 I took off the glove. By hole #9 I was used to the leather grips. By hole #18 I wasn’t thinking about the grips anymore.
At first they didn’t feel like they would provide me with a good grip on the club. I was completely wrong about that: I found that after a shot you have to practically peel your hands off of them. I was very surprised, and pleasantly so.