I have used Presta valve inner tubes for 40 years now. For the first 35 years I found Presta valves to be extremely reliable. The valve core would NEVER come loose during the life of the tube.

Unfortunately, for the past 5 years Presta valves have become extremely unreliable. 100% of valve cores now routinely come loose when I unscrew the top lock nut to pump the tire. I routinely have a sudden unexplained pressure loss of 10-20 psi per day, always caused by a loose Presta valve core.

I presume that Presta valve cores now fall apart easily because they are designed to come apart to add sealant of some kind. I understand that, but I don't use sealants. All I know is that I have extreme difficulty pumping the tires without the valve core coming loose.

I am seeking suggestions for fixing that problem. Should I use Loc-Tite on the valve core of every new tube?

I have had valve cores come loose in the past over a period longer than five years.

I just tighten them up with a pair of pliers, usually stay in place thereafter.

I also never had a problem with Presta tubes until last year.  It was very frustrating.  The only solution I could think of was the exact same one Mike used--tighten it up with a pair of needle-nosed pliers.  It worked. 

After getting frustrated by the same issue, I now only buy tubes with non-removable cores.  Kenda (a Chinese brand, I think) still seem to make them.

https://clikvalve.com/products/clik-valve-hi-flow-core-set-for-presta-stems-nickel-plated-brass-finish

Never tried these Clik valves but there are some good reviews.  It seems to require a pump adaptor. 

I've to have notice presta cores coming out recently. They can only be tightened so far before the rubber seal is crushed. I've tried Teflon plumbers tape with some success. 

One more vote for tightening the inner part with pliars. Pre-emptive action.

I had problems with a Lezyne mini pump repeatedly deflating tyres when detaching the pump. 

Lezyne pumps have a screw-on mechanism rather than a push-on to attach the pump to the valve. Mostly good, but not when the unscrewing also unscrews the inner part of presta valves. 

I have had this problem over last couple of years. Unscrew presta core a few threads, a touch of super glue from a pin or toothpick, tighten with pliers, this will do the trick!

I had a presta core come out when removing my Lezyne pump in 2019–I remember it well because I’d just installed a new tube after my first puncture ever on tour. I acquired a tiny valve core tool and it lives in the little tool bag on my bike. My current bike is set up tubeless so the little tool is now accompanied by a couple of spare valve cores.